core.ac.uk · 2020. 12. 16. · JHEP04(2020)047 Published for SISSA by Springer Received: February...
Transcript of core.ac.uk · 2020. 12. 16. · JHEP04(2020)047 Published for SISSA by Springer Received: February...
JHEP04(2020)047
Published for SISSA by Springer
Received: February 7, 2020
Revised: March 20, 2020
Accepted: March 24, 2020
Published: April 8, 2020
Twistor strings for N = 8 supergravity
David Skinner
Department of Applied Mathematics and Theoretical Physics,
Wilberforce Road, Cambridge CB3 0WA, U.K.
E-mail: [email protected]
Abstract: This paper presents a worldsheet theory describing holomorphic maps to
twistor space with N fermionic directions. The theory is anomaly free when N = 8.
Via the Penrose transform, the vertex operators correspond to an N = 8 Einstein super-
gravity multiplet. In the first instance, the theory describes gauged supergravity in AdS4.
Upon taking the flat space, ungauged limit, the complete classical S-matrix is recovered
from worldsheet correlation functions.
Keywords: Supergravity Models, Superstrings and Heterotic Strings, Topological Strings
ArXiv ePrint: 1301.0868
Open Access, c© The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP04(2020)047
JHEP04(2020)047
Contents
1 Introduction 1
2 The worldsheet supermanifold 6
2.1 Automorphisms 8
2.2 Deformations 9
3 The twistor string 10
3.1 Matter fields 10
3.1.1 The infinity twistor 11
3.1.2 The action 12
3.2 BRST transformations 14
3.2.1 The ghost multiplets 14
3.2.2 The BRST operator 16
3.3 Worldsheet anomaly cancellation 18
3.3.1 Zero modes 19
3.4 Vertex operators 21
3.4.1 The N = 8 supergravity multiplet 21
3.4.2 Picture changing operators 24
4 Scattering amplitudes in the flat space limit 25
4.1 A degenerate infinity twistor 26
4.2 The worldsheet Hodges matrix 27
4.2.1 Self-dual N = 8 supergravity 32
4.3 The conjugate Hodges matrix 32
4.4 The tree-level S-matrix 36
5 Discussion 36
5.1 The SL(2;C) system 36
5.2 Higher genus 37
5.3 Boundary correlation functions in AdS4 38
5.4 Other issues 40
A Some properties of algebraic βγ-systems 41
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1 Introduction
The most influential scattering amplitude in Yang-Mills theory is undoubtedly the Parke-
Taylor amplitude [1]
An,0 =〈i, j〉4 δ4(
∑i pi)
〈1, 2〉〈2, 3〉 · · · 〈n, 1〉. (1.1)
It describes the tree-level (colour-ordered) scattering of two gluons i and j of negative
helicity and n − 2 gluons of positive helicity, each of momentum pi = λiλi. Its talismanic
status rests on two pillars. First, its extraordinary simplicity assures that scattering am-
plitudes are far more managable objects than could be expected from momentum space
Feynman diagrams, encouraging us that their structures and properties will repay our close
attention. Second, it provides a remarkably fertile base for deeper explorations of the full
Yang-Mills S-matrix. In it, one already sees hints of the twistor action for Yang-Mills [2, 3]
and the associated MHV diagram formalism [4–7], the germ of the Grassmannian formu-
lation of all on-shell diagrams [8–10], and the amplitude’s factorization properties — a
crucial ingredient of BCFW recursion [11] — laid bare. Finally, and of particular relevance
to the present paper, (1.1) is an avatar of Witten’s representation [12]1
An,k =
∫d4(k+2)|4(k+2)Z
vol(GL(2;C))
1
(12)(23) · · · (n1)
n∏i=1
Ai(Z(σi)) (σidσi) (1.2)
of the n-particle NkMHV amplitude in N = 4 SYM as an integral over the space of degree
k + 1 rational curves in twistor space. Witten obtained this form by generalizing Nair’s
interpretation [13] of the Parke-Taylor amplitude in terms of (the leading trace part of) a
current correlator supported on a holomorphic twistor line.
This paper is concerned not with Yang-Mills theory, but with gravity. An expression
for all n-particle tree-level MHV amplitudes in gravity was found by Berends, Giele &
Kuijf [14] only two years after the discovery of the Parke-Taylor amplitude. Despite this, the
gravitational S-matrix has proved more resistant to study than its Yang-Mills counterpart.
Although gravity amplitudes admit a BCFW expansion [15–18], actually carrying it out
leads to expressions for n-particle NkMHV amplitudes whose structure is as yet unclear [19].
Applying Risager’s procedure to the BGK amplitude [20, 21] leads to an MHV diagram
formulation that fails when n ≥ 12 [22, 23], while the twistor action for gravity tentatively
proposed in [24] does not appear to extend naturally to N = 8 supergravity. Considering
that N = 8 supergravity is supposed to be the simplest quantum field theory, it has been
remarkably difficult to grapple with.
The situation changed dramatically with Hodges’ two papers [25, 26]. Hodges showed
that the tree-level MHV amplitude for gravity could be reformulated as
Mn,0 = 〈i, j〉8 det′(H) δ4
(∑i
pi
), (1.3)
1Throughout this paper, (ij) denotes the SL(2;C)-invariant inner product εαβσαi σ
β
j of the homogeneous
coordinates σα on an abstract curve Σ of genus zero. Z denotes a holomorphic map Z : Σ → PT to
N -extended supertwistor space, here with N = 4. In (1.2) this map has degree k + 1. Ai(Z) are twistor
representatives of the external wavefunctions. See [12] for further details.
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where H is the n× n symmetric matrix with entries
Hij =[i, j]
〈i, j〉for i 6= j, Hii = −
∑j 6=i
[i, j]
〈i, j〉〈a, j〉〈b, j〉〈a, i〉〈b, i〉
, (1.4)
where |a〉 and |b〉 are arbitrary spinors. The diagonal entries are nothing but the charac-
teristic gravitational ‘soft factors’ for the ith particle [14, 27, 28]. H has rank n − 3, and
det′(H) is any (n−3)×(n−3) minor of H, divided by the permutation symmetric combina-
tion 〈r1, r2〉〈r2, r3〉〈r3, r1〉 corresponding to the removed rows and also by a similar factor
〈c1, c2〉〈c2, c3〉〈c3, c1〉 for the removed columns. Hodges’ representation has many remark-
able properties. Chief among these is that Bose symmetry in the external states is achieved
through determinant identities rather than through an explicit sum over permutations.
Like the Parke-Taylor amplitude (1.1), (1.3) provides an inspirational starting point
from which to launch deeper investigations of the gravitational S-matrix. It opens up a path
by which to approach gravity on its own terms. In particular, unlike the BGK form, (1.3)
makes no mention of any cyclic ordering that is an artifact of trying to fit gravitational
pegs into a Yang-Mills hole. See [28, 29] for deconstructed forms of the Hodges amplitude
that were known previously, and [30, 31] for a graph-theoretic explanation of the relation
between them.
One outcome of these investigations was given in [32], where it was conjectured that
arbitrary n-particle NkMHV tree-level amplitudes in N = 8 supergravity could be repre-
sented as
Mn,k =
∫d4(k+2)|8(k+2)Z
vol(GL(2;C)) det′(H) det′(H∨)
n∏i=1
hi(Z(σi)) (σidσi) . (1.5)
This form was obtained by interpreting (1.3) in terms degree 1 holomorphic maps from a
Riemann sphere Σ into twistor space, and then generalizing to higher degree maps. Thus
it bears the same relation to (1.3) for N = 8 supergravity as (1.2) does to (1.1) for N = 4
SYM. In (1.5), H is the n× n matrix of operators
Hij =1
(ij)
[∂
∂µi,∂
∂µj
]for i 6= j, Hii = −
∑j 6=i
1
(ij)
[∂
∂µi,∂
∂µj
] k+2∏r=1
(arj)
(ari)(1.6)
that act on the external wavefunctions hi(Z), generalizing (1.4). Here, the ar are any k+ 2
points on Σ. The factor of det′(H) in (1.5) is any (n − k − 3) × (n − k − 3) minor of H,
divided by the Vandermonde determinant
|σr1 · · ·σrk+3| ≡
∏i<j
i,j∈removed
(rirj) (1.7)
of the worldsheet coordinates corresponding to the removed rows, and a similar factor for
the removed columns. The combination det′(H) is independent of the choice of minor. We
shall call H ‘the worldsheet Hodges matrix’, or often just ‘the Hodges matrix’. Similarly,
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JHEP04(2020)047
H∨ is the n× n matrix with elements
H∨lm =〈λ(σl), λ(σm)〉
(lm)for l 6= m,
H∨ll = −∑m 6=l
〈λ(σl), λ(σm)〉(lm)
n−k−2∏s=1
(asm)
(as l)
∏k 6=l,m
(k l)
(km), (1.8)
where the as are any n−k−2 points. The factor of det′(H∨) in (1.5) is any (k+1)× (k+1)
minor of H∨, divided by the Vandermonde determinant
|σl1 · · ·σlk+1| ≡
∏l<m
l,m∈remain
(rl rm) (1.9)
corresponding to the rows that remain in this minor, and again by a similar factor for the
remaining columns. Again, though it is not obvious from our current description, det′(H∨)
is completely permutation symmetric in all n sites. Under a parity transformation of
the amplitude, det′(H) and det′(H∨) are exchanged [33–35] (up to a Vandermonde factor
involving all n points), hence we shall call H∨ the ‘conjugate Hodges matrix’. When k = 0,
the Vandermonde determinant (1.9) should be taken to be unity and det′(H∨) itself is
almost trivial. This is why it is invisible in (1.3).
The conjecture that (1.5) correctly describes the complete classical S-matrix of N = 8
supergravity was proved (to a physicist’s level of rigour) in [33], where it was shown
that (1.5) obeys the twistor space form of BCFW recursion [36–39] at the heart of which
is the requirement that the amplitude has the correct behaviour in all factorization chan-
nels. (1.5) has also been shown to possess the correct soft limits [35]. For preliminary
investigations of a Grassmannian representation of (1.5), see [33, 34]. Using this Grass-
mannian at k = 0, an investigation of the MHV diagram formalism for gravity very re-
cently been carried out in [40]; excitingly, it has the potential to overcome the limitations
of Risager’s method. A different (presumably equivalent) generalization of Hodges’ form
to higher degree maps can be found in [41, 42].
The most striking property of the representation (1.5) is that it exists at all. The
unfathomable morass of Feynman diagrams that contribute to an n-particle gravitational
scattering process miraculously conspires to ensure that the tree amplitude lives on a
rational curve in twistor space! At MHV, this fact was originally seen by Witten in [12]
using the BGK form of the amplitude.2 It was also shown to hold for the 5-particle NMHV
amplitude in [43].3 The existence of (1.5) means, first and foremost, that all gravitational
tree amplitudes live on higher degree rational curves in twistor space.
Why should the gravitational S-matrix know about these curves? The answer pursued
here — really, the only conceivable answer — is that four-dimensional gravity is a twistor
string theory. The purpose of the current paper is to find this twistor string. Specifically,
2The derivatives in the Hodges matrix (4.2) are responsible for what was called ‘derivative of a δ-function
support’ in [12].3The non-trivial statement here is just that the 5-point NMHV amplitude has support on some CP2 ⊂ PT.
Any five points on a plane define a conic.
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JHEP04(2020)047
over the course of this paper we shall construct a worldsheet theory that localizes on
holomorphic maps to N = 8 supertwistor space and whose vertex operators correspond via
the Penrose transform to a linearized N = 8 supergravity multiplet. We shall obtain (1.5)
from worldsheet correlation functions of this theory at g = 0. This work thus provides
the theoretical framework in which (1.5) should be understood. See [31, 44–51] for earlier
attempts to understand Einstein gravity in the context of twistor strings.
It is clear that (1.5) possesses a rich and intricate structure. What clues are there to
guide us to the underlying theory? The main prompt follows from a simple counting that
also helped stimulate the discovery of (1.5). At g loops, n particle gravitational scattering
amplitudes are proportional to the κn+2g−2, where κ is the square root of the Newton
constant GN and has dimensions of (mass)−1. Since the amplitude itself is dimensionless,
these dimensions must be balanced by kinematic factors. But on twistor space, the only
object that fixes a mass scale is the infinity twistor — an antisymmetric bitwistor whose
presence breaks conformal invariance. With flat space-time, the infinity twistor appears
in two separate guises, corresponding to the 〈 , 〉 and [ , ] brackets familiar from spinor
momenta. With n particles at g loops, the twistor space amplitude needs to contain
precisely n+ 2g− 2 factors of 〈 , 〉 and [ , ] in total. Under a parity transformation 〈 , 〉 and
[ , ] are exchanged, along with the numbers n± of positive and negative helicity gravitons
participating in the scattering process (in the pure gravity sector). We deduce that the
twistor space amplitude must be proportional to
n+ + g− 1 = n− k + g− 3 factors of [ , ] and
n− + g− 1 = k + g + 1 factors of 〈 , 〉 .(1.10)
Note that the symmetric choice (n + 2g − 2)/2 is not possible since n may be odd and
the integrand is rational. The fact that n+ and n− respectively go with [ , ] and 〈 , 〉 is a
convention fixed by comparison with (1.3). This dependence is certainly realized in (1.5),
where det′(H) is easily seen to be a monomial of degree n−k−3 in [ , ] whereas det′(H∨) is a
monomial of degree k+1 in 〈 , 〉. At higher loops, (1.10) is compatible with all factorization
channels of g-loop NkMHV amplitudes.
The key question is to ask what sort of worldsheet objects could be responsible for
this behaviour. For [ , ], the answer we find is that k − g + 3 of the vertex operators are
fixed and do not involve the infinity twistor, whereas the remaining n − k + g − 3 are
integrated and are linear in [ , ]. To achieve this, the worldsheet theory will involve a field
with (generically) k − g + 3 zero modes whose fixing is associated to the Vandermonde
factor (1.7) in det′(H). The dependence on [ , ] in the integrated operators is introduced
by the BRST operator responsible for the descent procedure. The infinity twistor in the
form [ , ] really endows twistor space with a twisted holomorphic Poisson structure, and
the BRST operator we arrive at is somewhat reminiscent of those found in Poisson sigma
models [52, 53].
The theory also contains a different field with (generically) k+g+ 1 zero modes whose
fixing provides the Vandermonde determinant (1.9) in det′(H∨). The vertex operators as-
sociated to this field have a rather different character that may be motivated as follows.
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JHEP04(2020)047
Firstly, notice that the previous integrated and unintegrated vertex operators have appar-
ently already used up all the available n insertion points. (1.5) allows us to choose the
minors of H and H∨ completely independently, so there seems to be no compelling reason
to place the ‘additional’ operators preferentially with either type of insertion mentioned in
the previous paragraph. Secondly, although det′(H∨) knows about the holomorphic map
Z : Σ→ PT, it is completely oblivious to the external world. Unlike det′(H) which knows
about the external states through the derivative operators in (1.6), nothing in the definition
of det′(H∨) cares what we choose for the wavefunctions hi(Z), nor even how many particles
are being scattered. All this is strongly reminiscent of ‘picture changing operators’ of the
RNS superstring. See e.g. [54, 55] for an introduction to these operators. We shall indeed
find picture changing operators in our theory, and inserting k + g + 1 of these leads to the
requisite dependence on 〈 , 〉.The characterization sketched above may sound worryingly piecemeal. On the con-
trary, the whole theory flows naturally from a single structure: the worldsheet carries a
certain exotic twisted supersymmetry. All the required objects fit into geometrically mean-
ingful worldsheet supermultiplets — properly understood, the theory contains only three
different fields. The action and BRST operator are as simple as one could wish.
The outline of the paper is as follows. In section 2 we describe the worldsheet super-
manifold whose fermionic symmetries and moduli lie at the heart of the whole construction.
The actual worldsheet theory is a relative of Berkovits’ formulation [56] of the original
twistor string, and is described in section 3. (Some readers may prefer to begin with this
section.) Here we begin with a description of the worldsheet fields in section 3.1 and BRST
operator in section 3.2. After a brief diversion, we proceed to show in section 3.3 that the
model is anomaly free if and only if the target twistor space has N = 8 supersymmetry. We
conclude our discussion of the general theory in section 3.4, presenting the vertex operators
of the model and explaining their relation to an N = 8 supergravity multiplet. Section 4
contains the derivation of the complete flat space S-matrix of classical N = 8 supergrav-
ity (1.5) from correlation functions of vertex operators on the worldsheet. The model of
section 3 describes SO(8) gauged supergravity on an AdS4 background in the first instance.
Thus, before embarking on the S-matrix calculation, in section 4.1 we show in section 4.1
how to rescale the fields so as to take the flat space limit. The Hodges matrix (1.6) and
the conjugate Hodges matrix (1.8) have a different origin on the worldsheet. They are
obtained in sections 4.2 and 4.3, respectively. It is worth pointing out immediately that
the present model is more successful than the original twistor strings [12, 56] were (as a
theory of pure N = 4 SYM) in at least one respect: the worldsheet correlator we consider
leads inexorably to (1.5) and only to (1.5). No terms are ignored or discarded by hand.
Our work suggests many promising avenues for future research. We conclude in section 5
with a brief discussion of some of these.
Note added at publication. In the time since this paper first appeared on the arXiv,
there have been several related developments. The most important of these is the emer-
gence of the CHY representation of amplitudes [57, 58] and its realization as a string theory
in ambitwistor space [59]. The ambitwistor string appears to be a more flexible framework
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JHEP04(2020)047
than that presented here, in particular for amplitudes in various theories [60, 61] in var-
ious dimensions, loop-level amplitudes [62–64, 66, 67] and for describing amplitudes on
curved backgrounds [68, 69]. (For progress on higher genus twistor strings, see [70].) On
the other hand, the twistor string allows for manifestly supersymmetric expressions for
amplitudes that are obscured by the usual CHY formulae. The precise relation between
the ambitwistor and twistor strings has not yet emerged, though ambitwistor versions of
the twistor string [71] and the ‘polarized scattering equations’ of [72–74] make it clear that
they are indeed very closely related. It seems likely that the relation will be somewhat
similar to that between the RNS and pure spinor versions of full string theory [75].
2 The worldsheet supermanifold
In this section we describe the geometry of the worldsheet supermanifold X on which the
twistor string theory is based. See e.g. [76–78] for much more information on complex
supermanifolds.
Let Σ be a closed, compact Riemann surface of genus g. We extend Σ to a complex
supermanifold X of dimension 1|2 by picking4 a line bundle L → Σ of degree d ≥ 0 and a
choice of spin bundle K1/2Σ . We then define X to be the split supermanifold whose tangent
bundle TX is
TX = TΣ⊕D , (2.1)
where D is the rank 2 fermionic bundle5
D ∼= Π(C2 ⊗K−1/2
Σ ⊗ L)
(2.2)
over Σ. We will often say that objects taking values in KpΣ⊗L
q have spin p and charge q.
Thus, sections of D have spin −12 and charge +1.
For a local description of X, we cover the bosonic Riemann surface Σ by open coordinate
patches Uα and let Uα be the corresponding cover of X. Each such Uα is (an open
subset of) C1|2, so we may describe X locally in terms of one bosonic and two fermionic
holomorphic coordinates z|θa, with a = 1, 2. The fact that X is a split supermanifold
means that on overlaps the coordinate transformations are
zα = fαβ(zβ)
θaα = (gαβ(zβ))ab θbβ ,
(2.3)
where the transition functions fαβ and gαβ on Uα ∩ Uβ depend only on the bosonic coor-
dinate z, not (z|θ). To identify these transition functions, suppose we write an arbitrary
section V : X→ TX of the tangent bundle of X as
V = V z(z|θ) ∂∂z
+ Va(z|θ) ∂
∂θa. (2.4)
4Eventually, the twistor string path integral will include a sum (or integral) over all such choices.5Here, Π is the ‘parity reversing functor’ whose role is simply to remind us that D is fermionic.
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JHEP04(2020)047
Recalling that TX ∼= TΣ⊕D, we see that V z∂z is a section of TΣ (written in terms of the
local basis ∂/∂z) whilst Va∂a is a section of D (written in terms of the local basis ∂/∂θa).
In order to compensate the transformations of the basis
∂
∂zα=
1
f ′αβ
∂
∂zβand
∂
∂θaα= (g−1
αβ ) ba
∂
∂θbβ(2.5)
that follow from (2.3), the components V z and Va must transform as
V z(zα|θα) = f ′αβ Vz(zβ |θβ) and Va(zα|θα) = (gαβ)ab V
b(zβ |θβ) (2.6)
on overlaps. But since D ∼= C0|2 ⊗ T 1/2Σ ⊗ L, we have
(gαβ)ab =√f ′αβ × (hαβ)ab (2.7)
where each component of the 2× 2 matrix h is a transition function for sections of L.
Because X is a split supermanifold, it can be viewed as the total space of a bundle over
Σ — indeed, this is just what is meant by the transformation laws (2.3). To identify this
bundle, note that by (2.7) and (2.6), the coordinates θ themselves transform as components
of a section of D. But since the coordinates on a bundle transform oppositely to the bundle
itself, we find that X is the total space of D∨ → Σ, where D∨ is the dual of D. To say
this differently, functions on X are superfields Φ(z|θ) that may be expanded in the usual
way as
Φ(z|θ) = φ(z) + θaψa(z) +1
2εabθ
aθbξ(z) (2.8)
where φ(z) is a function on Σ, the ψa are a pair of functions on Σ with values in K+1/2Σ ⊗L−1
and of opposite Grassmann parity to φ, and ξ is a function on Σ with values in KΣ⊗L−2.
Thus, the structure sheaf of X is OX = OΣ(∧∗D∨). More generally, if Φ[p,q](z|θ) is a section
of KpΣ⊗L
q , then it may be expanded in terms of fields φ[p,q], ψ[p,q]a and ξ[p,q] on Σ, of (spin,
charge) = (p, q), (p+ 12 , q − 1) and (p+ 1, q − 2), respectively.
To give an example that will be important later, suppose that Σ is the Riemann
sphere. On CP1, the bundles K−1/2 and L are uniquely determined to be O(1) and O(d),
respectively. Thus D = C0|2 ⊗O(d + 1) and the supermanifold X may be identified as the
weighted projective superspace WCP(1,1|d+1,d+1) with homogeneous coordinates (σα|ϑa)obeying the scaling
(σα|ϑa) ≡ (rσα|rd+1 ϑa) for any r ∈ C∗ . (2.9)
In this case, a function Φ ∈ OX may be expanded as
Φ(σ|ϑ) = φ(σ) + ϑaψa(σ) +1
2εabϑ
aϑbξ(σ) , (2.10)
where φ, ψa and ξ have homogeneities 0, −(d + 1) and −2(d + 1), respectively under the
scaling (2.9).
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JHEP04(2020)047
Returning to the general case, the cotangent bundle T∨X to the supermanifold is just
the direct sum T∨X ∼= KΣ ⊕ D∨ dual to (2.1). Thus the holomorphic Berezinian Ber(X)
of X is
Ber(X) = Ber(KΣ)⊗ Ber(D∨) . (2.11)
To compute this, recall that for an even parity (bosonic) bundle Ber(B) = Det(B) —
the top exterior power of B. However, for an odd parity (fermionic) bundle Ber(F ) =
Det(ΠF )−1, where the power of −1 appears as a consequence of the fact that in Berezin
integration the integral form dθ1dθ2 transforms oppositely to the differential form dζ1∧dζ2
involving variables ζa that have same quantum numbers, but opposite Grassmann parity to
θa. (In one dimension, this is just the familiar statement that since∫
dθ θ = 1 by Berezin
integration, if θ → g(z)θ, we require that the integral form dθ → g−1(z)dθ.) Thus we have
Ber(X) = KΣ ⊗Det(ΠD∨)−1 = KΣ ⊗(KΣ ⊗ L−2
)−1
∼= L2 ,(2.12)
where we used the definition (2.2) in the second step.
When we come to write the worldsheet action in section 3, we will need a top holo-
morphic integral form on X. Since the Berezinian of X is isomorphic to L2, the product
Ber(X)⊗L−2 is trivial. Thus it admits a global holomorphic section that we write as d1|2z.
For example, at genus zero d1|2z = (σdσ)dϑ1dϑ2 in terms of the homogeneous coordinates
(σα|ϑa) introduced above. We can treat d1|2z as a top holomorphic (integral) form on X of
charge −2. In order to construct an action, this charge must be balanced by the worldsheet
Lagrangian L, so that d1|2z L(z|θ) may be integrated over X.
Let us close this subsection with a couple of remarks. As usual for complex superman-
ifolds (and as on the twistor target space CP3|N ) we take X to be a cs manifold [76, 77], in
the sense that the antiholomorphic tangent bundle is TX ≡ TΣ and so has rank 1|0. An-
tiholomorphic fermionic directions simply do not exist — all operations with the fermions
will be purely algebraic. Finally, we note that X is not an N = 2 super Riemann surface
(see e.g. [78]), because our choice of TX means the distribution D is integrable in the sense
that D,D ⊆ D. Indeed, the usual superderivatives
D1 =∂
∂θ1+ θ2 ∂
∂zD2 =
∂
∂θ2+ θ1 ∂
∂z(2.13)
on an N = 2 super Riemann surface do not make any sense for us, because the second
term in each expression has different charge from the first and hence is forbidden. Exactly
these forbidden terms are responsible for the non-integrability of the odd distribution on
an N = 2 super Riemann surface.
2.1 Automorphisms
We now consider the symmetries of X as a complex supermanifold. On a local patch Uα,
as usual these are generated by vector fields V ∈ Ω0(Uα, TX|Uα
). We will be particularly
interested in the symmetries of the distribution D — these are generated by the vector
fields V ∈ Ω0(Uα, D|Uα) that act trivially on Σ. (This restriction would not make sense on
a super Riemann surface, but does make sense on X precisely because D is integrable.)
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JHEP04(2020)047
From (2.4) we can write
Va(z|θ) ∂
∂θa=
(va(z) + θbRab(z) +
1
2εbcθ
bθc va(z)
)∂
∂θa(2.14)
where the components v, R and v have (spin, charge) = (−12 , 1), (0, 0) and (+1
2 ,−1), re-
spectively. Because the fermionic distribution D is integrable, the anticommutator V1,V2of any two such vector fields again lies in D. A short calculation shows that the component
fields obey the algebra
[v1, R2] = v12 [v1, R2] = v12
[R1, R2] = R12 v1, v2 = R′12 ,(2.15)
where
va12 = (R2)abvb1 va12 = (R2)abv
b1 − tr(R2)va1
(R12)ab = (R2R1 −R1R2)ab (R′12)ab = −va2 v1b .(2.16)
with vb = εbcvc. All other commutators are zero — in particular, v1, v2 = 0 since it is of
order (θ)3 which must vanish.
If we decompose the gl(2;C) matrix R as
Rab =1
2δab r + rab (2.17)
where the traceless, symmetric matrix rab takes values in sl(2;C) while r = tr(R) takes
values in gl(1;C). R may be interpreted as generating a gauge transform of C2⊗L so that r
generates gauge transformations associated to the determinant L2. Equations (2.15)–(2.16)
then reflect the fact that the va transform in the fundamental representation of SL(2;C)
and have charge +1 under L, whereas the va transform in the fundamental of SL(2;C) but
have charge −1 under L.
Later, in section 3.2 we shall introduce a (bosonic) ghost multiplet in ΠΩ0(X,D)
corresponding to (2.14). This algebra will then be interpreted as the gauge algebra of our
worldsheet theory. Zero modes of the ghost multiplet live in H0(X,D), parity reversed, and
correspond to globally defined Σ-preserving infinitesimal automorphisms of X as a complex
supermanifold.
2.2 Deformations
As for a usual complex manifold, infinitesimal deformations of X as a complex supermani-
fold are parametrized by elements of H1(X, TX). This cohomology group is the holomorphic
tangent space to the moduli space of X. Again, we will be interested in the moduli of X
associated to deforming the choice of distribution D whilst leaving Σ fixed. Infinitesimally,
these are described by H1(X,D) ⊂ H1(X, TX). On a supermanifold, the dualizing sheaf is
the holomorphic Berezinian, so the deformations are Serre dual6 to H0(X,Ber(X) ⊗ D∨).
6See e.g. [78–80] for a discussion of Serre duality for complex supermanifolds.
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We showed in (2.12) that Ber(X) ∼= L2, so using (2.2) we can identify this cohomology
group as
H0(X,Ber(X)⊗D∨) ∼= ΠH0(X,C2 ⊗K1/2Σ ⊗ L) , (2.18)
where again the symbol Π denotes that the fibres are fermionic. This group is non-trivial
provided only deg(L) > 0, so X has odd moduli even at g = 0. In section 3.2 we shall
introduce a (bosonic) antighost multiplet valued in Ω0(X,C2 ⊗K1/2Σ ⊗ L). Zero modes of
this antighost live in (2.18), parity reversed, and so by Serre duality can be paired with
deformations of the odd moduli. This is the usual mechanism by which the RNS superstring
provides a top holomorphic integral form on odd moduli space; see e.g. [55, 78].
3 The twistor string
In this section we define the worldsheet theory that will provide a twistor description of
Einstein supergravity. After introducing the fields and explaining their geometric meaning,
we study the gauge and BRST transformations naturally associated to the structure of the
worldsheet supermanifold X. The model is chiral, and we show that all (local) worldsheet
anomalies vanish if and only if the target space has N = 8 supersymmetry. We then
construct vertex operators in the BRST cohomology, finding that they correspond to an
N = 8 supergravity multiplet. We have just seen that X has odd moduli even at genus
zero. We construct the associated ‘picture changing’ operators.
3.1 Matter fields
To define the worldsheet model, we first introduce four bosonic and N fermionic fields ZI
(where I = 1, . . . , 4|1, . . . ,N ). Each of these are scalars on X of charge +1. In other words,
Z ∈ Ω0(X,C4|N ⊗ L) (3.1)
where L is the same degree d line bundle used in the definition (2.2) of D. In the first
instance, Z defines a smooth map Z : X → C4|4. The twisting by L means that this
map is defined only up to an overall non-zero complex rescaling, so Z really defines a map
Z : X→ CP3|N . The ZI then represent the pullbacks to X of homogeneous coordinates on
this projective space.
Saying that Z is a map from X, rather than from Σ, simply means that it is a worldsheet
superfield. As in (2.8), we define its component expansion to be
ZI(z, θ) = ZI(z) + θaρIa(z) +1
2θaθa Y
I(z) (3.2)
in terms of fields (ZI , ρIa, YI) on Σ. Since each θa is a fermionic coordinate of (spin,
charge) = (−12 , 1) we see that ZI , like ZI , is a scalar on Σ of charge +1, ρIa are a pair
of uncharged (12 , 0)-forms, while Y I is a (1,0)-form of charge −1. If the index I denotes
a bosonic direction in CP3|N , then ZI and Y I are bosons while ρIa are fermions. This is
reversed when I denotes a fermionic direction. Altogether, the component fields in the
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matter multiplet areZ ∈ Ω0(Σ,C4|N ⊗ L)
ρ ∈ ΠΩ0(Σ,C4|N ⊗ C2 ⊗K1/2Σ )
Y ∈ Ω0(Σ,C4|N ⊗KΣ ⊗ L−1) .
(3.3)
Of course, the ZI represent the pullbacks, now to Σ, of homogeneous coordinates of CP3|N .
We will sometimes decompose ZI as ZI = (Za|χA) = (µα, λα|χA) into its bosonic and
fermionic components.
3.1.1 The infinity twistor
In order to write an action for these fields, we must pick some extra data. This is a choice
of constant, graded skew symmetric bi-twistor IIJ , known as the ‘infinity twistor’. Graded
skew-symmetry means that IIJ = −(−1)|IJ | IJI , where |IJ | = 1 if both I and J denote
fermionic directions, and zero otherwise. For any two twistors Z1 and Z2, we will usually
denote IIJZI1ZJ2 by 〈Z1, Z2〉.Projective twistor space carries a natural action of SL(4|N ;C) acting as linear trans-
formations on the homogeneous coordinates. This is the complexification of (the double
cover of) the space-time N -extended superconformal group. The role of the infinity twistor
is to break conformal invariance and determine a preferred metric on space-time [81–83].
Specifically, if Xab = Z[a1 Z
b]2 are homogeneous coordinates for the bosonic part of the twistor
line Z1Z2, representing a point x in space-time, then
ds2 =εabcddXabdXcd
(IefXef)2(3.4)
is the space-time metric. According to this metric, lines in twistor space that obey I ·X = 0
lie ‘at infinity’ in space-time. The fermionic-fermionic components IAB were examined
in [84, 85] where it was shown that they likewise define a metric on the R-symmetry group.
A non-trivial IAB thus corresponds to gauging the R-symmetry.
For definiteness, we will make the choice
IIJ =
(Iab 0
0 IAB
)(3.5)
where the even-even components Iab and odd-odd components IAB are given by7
Iab =
(Λεαβ 0
0 εαβ
)and IAB =
√Λ δAB , (3.6)
respectively. This infinity twistor is non-degenerate. Its inverse is IIJ/Λ, where
IIJ =
εαβ 0 0
0 Λεαβ 0
0 0√
Λ δAB
. (3.7)
IIJ defines a holomorphic Poisson structure IIJ ∂∂ZI∧ ∂
∂ZJof homogeneity −2 on twistor
space. It will play an important role in the vertex operators.
7The fact that IIJ with lower indices contains εαβ with raised indices originates with our conventions
that the bosonic twistor components Za are written as (µα, λα).
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In (3.6)–(3.7), Λ is a constant of dimensions (mass)2. The powers of Λ can be un-
derstood as follows. Since x has dimensions (mass)−1 and λ has dimensions (mass)12 , the
incidence relations µα = xααλα show that µ has dimensions (mass)−12 . Similarly, the
space-time fermionic coordinate ϑ has dimensions (mass)−12 , so the twistor space fermionic
directions χ are dimensionless. The powers of Λ ensure that both
IIJZIdZJ = Λεαβµαdµβ + εαβλαdλβ +
√Λ δAB χ
AdχB (3.8)
and the Poisson structure
IIJ ∂
∂ZI∧ ∂
∂ZJ= εαβ
∂
∂µα∧ ∂
∂µβ+ Λεαβ
∂
∂λα∧ ∂
∂λβ+√
Λ δAB∂
∂χA ∂
∂χB(3.9)
have homogeneous dimension (mass)+1. Bosonically at least, this dimension is important
in ensuring that (3.4) indeed has dimensions (mass)−2 as expected for a space-time metric.
Recall that the n-particle g-loop gravitational scattering amplitude comes with a factor of
κ2g−2+n. These dimensions must be balanced by a total of 2g− 2 + n powers of I.
With the choice (3.6), the incidence relations show that the space-time metric (3.4)
becomes
ds2 =ηµνdxµdxν
(1 + Λx2)2(3.10)
where ηµν is the flat metric. This is the metric of (complexified) AdS4 with cosmologi-
cal constant Λ. Similarly, with IAB =√
Λ δAB the SL(N ;C) R-symmetry is broken to
SO(N ;C). Thus, with the choice (3.6) of infinity twistor, our model will describe (subject
to an appropriate reality condition) SO(N ) gauged supergravity on an AdS4 background.
It is straightforward to introduce an arbitrary gauge coupling for the gauged R-symmetry
by rescaling IAB → g√
Λ δAB for some dimensionless coupling g. In section (4.1) we shall
take the limit Λ→ 0 (with g remaining fixed) so as to compute the flat space S-matrix of
ungauged supergravity. Until then, we set Λ = 1 and g = 1 so as to lighten the notation.
3.1.2 The action
Having chosen our infinity twistor, the action for Z is simple to state. We have8
S1 =1
4π
∫X
d1|2z 〈Z, ∂Z〉
=1
2π
∫Σ〈Y, ∂Z〉 − 1
2〈ρa, ∂ρa〉 ,
(3.11)
where ρaI = εabρIb . Notice that the charge +2 of 〈Z, ∂Z〉 balances the charge −2 of d1|2z. In
writing this action, we let ∂ denote the (covariant) Dolbeault operator acting on sections of
the appropriate bundles — since the (0,2)-part F 0,2 of the curvature of any bundle vanishes
trivially on restriction to a Riemann surface, we can always work in a gauge in which the
(0,1)-form part of any connection is the usual ∂-operator. (This applies equally to the cs
manifold X.)
8In computing the component expansion of this action, we use the conventions∫
d2θ θaθa = 2 where
θaθa = εabθaθb. Note also that θaθb = − 1
2εab θcθc.
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Clearly, to have sensible kinetic terms for all components of Z, it is important that the
infinity twistor 〈 , 〉 be totally non-degenerate, both in the bosonic directions and fermionic
directions. This motivates our choice (3.6). When we come to take the flat space limit in
section 4 so as to describe scattering amplitudes, the infinity twistor necessarily becomes
degenerate. We shall then need to rescale the fields so as to remove the dependence on the
cosmological constant from the action, at the cost of including it in the definition of the
Z(z, θ) supermultiplet.
The Y Z-system9
SY Z =1
2π
∫ΣYI∂Z
I (3.12)
was a key ingredient of Berkovits’ twistor string [56]. Here, as there, performing the path
integral over Y will lead to the constraint that Z be a holomorphic section of C4|N ⊗ L.
Thus, on-shell, Z describes a holomorphic map to PT. Berkovits’ twistor string describes
non-minimal N = 4 conformal supergravity [86–88], at least at tree-level, as does Witten’s
original model [12]. The N = 4 (conformal) gravity multiplet is not self-conjugate un-
der CPT transformations and to build a CPT invariant theory we must use two separate
supermultiplets that are exchanged under CPT. These two multiplets have a very differ-
ent character on twistor space. The multiplet containing the positive helicity graviton is
described locally by a vector field V , whereas the multiplet that contains the negative he-
licity graviton is instead described locally by a 1-form B.10 Correspondingly, in Berkovits’
twistor string the two conformal gravity multiplets are represented by the worldsheet vertex
operators V I(Z)YI and BI(Z) dZI , respectively [87].
Conformal gravity, being a fourth order theory, contains twice as many on-shell degrees
of freedom as Einstein gravity. If we wish to extract the Einstein supergravity multiplets
from the vertex operators of the Berkovits twistor string, we should require that
V IYI = (IIJ∂Jh)YI and B = φ 〈Z, dZ〉 (3.13)
for some (local) functions h(Z) and φ(Z) of homogeneities +2 and −2, respectively.
(See [24, 81, 83, 85] — or section 3.4 below — for further details.) One of the challenges
to be overcome in constructing a twistor string for N = 8 supergravity is to understand
how to unify these ‘vector field’ and ‘one form’ vertex operators as part of a single CPT
self-conjugate N = 8 multiplet.
Although it is premature to discuss the spectrum of our model at this point, (3.2)
already contains a small hint of the solution: the fields Z and Y are unified into a sin-
gle worldsheet supermultiplet. Thus, from the perspective of X, there is no fundamental
difference between the two types of vertex operator in (3.13).
9Here we have used the infinity twistor I to lower the index on Y , so that YI = IIJY J . With a non-
degenerate infinity twistor, this is harmless, but in the flat space case this operation must be done with
care. See section 4.1.10More precisely [87], in N = 4 conformal supergravity V represents an element of H1(PT, TPT) and
defines an infinitesimal deformation of the complex structure of a patch of twistor space that preserves the
holomorphic section D3|4Z of Ber(PT). The conjugate field B plays a role similar to that of the heterotic
B-field. Its curvature H = dB represents an element of H1(PT,Ω2cl).
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3.2 BRST transformations
To complete the specification of our theory, we need to choose a BRST operator. This will
be based on the symmetries of X that act trivially on Σ, as discussed in section 2.1.
Consider the following three sets of transformations of the component fields. Firstly,
δ1ZI = εaρIa , δ1ρ
Ia = εaY
I , δ1YI = 0 (3.14)
with fermionic parameters εa, secondly
δ2ZI =
1
2κ aa Z
I , δ2ρIa = −κ b
a ρ , δ2YI = −1
2κaaY
I (3.15)
with bosonic parameters κ ba , and finally
δ3ZI = 0 , δ3ρ
Ia = −1
2εaZ
I , δ3YI =
1
2εaρIa (3.16)
with fermionic parameters εa. These transformations represent the actions of a local sym-
metry of X on the matter multiplet Z. On a local coordinate patch U ⊂ Σ, we may
take the parameters (ε, κ, ε) to be constant. In this case the action S1 is invariant un-
der (3.14)–(3.16) when restricted to U . However, the spins and charges of the compo-
nent fields mean that if we wish to make sense of these transformations globally over
Σ, then ε and ε must transform non-trivially on overlaps. Specifically, we must have
εa ∈ ΠΩ0(Σ,L⊗K−1/2Σ ), κab ∈ Ω0(Σ,O) and εa ∈ ΠΩ0(Σ,L−1⊗K+1/2
Σ ), so that they fit
together to form a supermultiplet in Ω0(X,D). In particular, to treat (3.14)–(3.16) globally
over Σ, the parameters ε and ε must depend (smoothly) on the worldsheet coordinates.
Thus these transformations must necessarily be gauged.
3.2.1 The ghost multiplets
With non-constant parameters, the matter action is not invariant under (3.14)–(3.16).
To remedy this, and to treat the transformations as redundancies, we follow the usual
procedure of introducing ghosts. Since the above transformations reflect the actions of
Σ-preserving symmetries of X as studied in section 2.1, we introduce a ghost multiplet
C ∈ ΠΩ0(X,D) (3.17)
in the parity reverse of the parameter multiplet. We declare C to have ghost number
ngh = +1. As with the matter field, we can expand C in terms of components as
Ca(z, θ) = γa(z) + θb Nab(z) +
1
2θbθb ν
a(z) , (3.18)
where again a = 1, 2. Recalling from section 2 that ΠD ∼= C2 ⊗K−1/2Σ ⊗ L, we see that
the component fields γa are a pair of are (−12 , 0)-forms of charge +1. They are bosonic
ghosts for the fermionic parameters εa in (3.14). The bosonic fields νa are likewise (+ 12 , 0)-
forms of charge −1 and are ghosts for fermionic parameters εa in (3.16). Finally, N ba are
four fermionic ghosts corresponding to the GL(2;C) transformations of the rank 2 bundle
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JHEP04(2020)047
C2 ⊗ L. They are scalars on Σ of charge 0. We shall often find it convenient to separate
this GL(2;C) as GL(1;C)× SL(2;C). Accordingly, as in (2.17) we write
Nab =
1
2δab n + nab (3.19)
where nab is symmetric and traceless and n ≡ tr(N). Note that since n is the (anticommut-
ing) gauge parameter for the determinant of C2 ⊗L, the appropriate parameter for gauge
transformations of L itself is n/2. This explains various factors of 12 that appear in the
BRST transformations below.
We also introduce an antighost multiplet
B ∈ ΠΩ0(X,Ber(X)⊗D∨) (3.20)
of ngh = −1 that is conjugate to C. This may be expanded as
Ba(z, θ) = µa(z) + θb Mab(z) +1
2θbθb βa(z) . (3.21)
Since ΠBer(X) ⊗ D∨ ∼= C2 ⊗K+1/2Σ ⊗ L, we see that the two bosonic fields µa are (1
2 , 0)-
forms of charge +1, the fermionic antighosts Mab are uncharged (1, 0)-forms, and finally
βa are a pair of bosonic ( 32 , 0)-forms of charge −1. As with the ghost N, we shall often
separate the antighost M into its GL(1;C) and SL(2;C) parts, writing
Mab = εab m + mab (3.22)
with mab symmetric and traceless.
The ghost action is simply
S2 =1
2π
∫X
d1|2z Ba∂Ca
=1
2π
∫Σβa∂γ
a + mab∂nab + m∂n + µa∂νa .
(3.23)
Except for their non-trivial charges under L, the βγ-system is just two copies of the usual
βγ system of the RNS superstring, the MN-system is the standard system associated to
fixing the GL(2;C) symmetry of D, and the µν-systems corresponds to fixing supergauge
transformations associated to gauginos. The non-trivial charges of these ghost fields mean
that the gravitinos and gauginos that they fix are also charged.
The above behaviour is perhaps reminiscent of a GL(2;C) gauged supergravity on the
worldsheet. However, because we are only gauging those symmetries of X that act trivially
on the bosonic Riemann surface Σ, we are not actually considering worldsheet gravity itself.
Correspondingly, our ghosts live only in the subgroup ΠΩ0(X,D) of ΠΩ0(X, TX) and there
is no (fundamental) fermionic bc-ghost system. It may seem strange to have gravitinos
(albeit non-propagating ones supplanted by the βγ-system) but no graviton. Usually in
supersymmetry, this is not allowed because the structure of the supersymmetry algebra
Q,Q† = P forces us to gauge Poincare transformations if we gauge the supersymmetry.
In the present case it is possible to have gravitinos without gravitons ultimately because
the distribution D is integrable and D,D 6⊃ TΣ.
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3.2.2 The BRST operator
In the presence of the ghosts, the transformations (3.14)–(3.16) are replaced by BRST
transformations generated by the operator
Q =1
2
∮d1|2z (Ca〈Z, DaZ〉+BaC,Ca) , (3.24)
where the derivative Da ≡ ∂/∂θa and where C,Ca = 2CbDbCa denotes the anticommu-
tator — i.e., the graded Lie bracket on the distribution D. The integral is to be taken over
a real 1-dimensional cycle Γ ⊂ Σ as well as over the fermonic directions. Clearly, Q is a
fermionic operator of ngh = +1, and the spins and charges of the fields and measure d1|2z
combine to ensure that Q is a scalar of charge zero under GL(2;C). It is also important to
notice that the BRST operator depends on our choice of infinity twistor 〈 , 〉.Performing the integrals over the anticommuting coordinates θa, (3.24) may equiva-
lently be written as11
Q =
∮γa〈Y, ρa〉+
1
2νa〈Z, ρa〉+
n
2〈Y, Z〉 − 1
2nab〈ρa, ρb〉
+ βa
(n
2γa + nabγ
b
)+ µa
(− n
2νa + nabν
b
)+ mγaνa −mab
(n(a
cnb)c + γ(aνb)
)(3.25)
in terms of the component fields (3.2), (3.18) & (3.21). When acting on the matter multiplet
Z(z, θ), this operator generates the transformations
δZI = γaρIa +n
2ZI
δρIa = γaYI +
1
2νaZ
I − n ba ρ
Ib
δY I =1
2νaρIa −
n
2Y I
(3.26)
generalizing (3.14)–(3.16). Similarly, the BRST transformations act as
δγa =n
2γa + nabγ
b δνa = −n
2νa + nabν
a
δn = νaγa δnab = n(acnb)c + γ(aνb)
(3.27)
on the ghost multiplet. These transformations directly reflect the structure of the alge-
bra (2.15)–(2.16). For example, we see that γa transform in the fundamental of SL(2;C)
and have charge +1 under L, while νa is again in the fundamental of SL(2;C) but has
charge −1. The unusual factors of νγ in the transformation of n and nab come from the
fact that v1, v2 = R′12 in (2.15)–(2.16). Finally, the BRST transformations act on the
11Our conventions are that f (agb) is the symmetrized product 12(fagb + f bga). Two-component indices
a, b, . . . , are raised and lowered using the SL(2)-invariant anitsymmetric tensor εab. In particular, for bosonic
fields such as γ and ν, γaνa = εabγaνb = −εbaνbγa = −νbγb. Pairs of anticommuting fields would have an
extra minus sign. Some care has been taken to ensure the numerical factors in (3.25)–(3.28) are correct
and compatible with the numerical factors in the matter and ghost action.
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JHEP04(2020)047
antighost multiplet as
δµa =1
2〈ρa, Z〉+
n
2µa − n b
a µb + mγa + mabγb
δm =1
2(〈Z, Y 〉 − βaγa + µaν
a)
δmab =1
2〈ρa, ρb〉 − 2n c
(a mb)c + β(aγb) + µ(aνb)
δβa = 〈ρa, Y 〉 −1
2nβa − n b
a βb −mνa + mabνb ,
(3.28)
giving the currents conjugate to each symmetry. The transformations (3.26)–(3.28) have
been checked to be nilpotent and to be symmetries of the full action S1 +S2. Of course, the
statement that Q2 = 0 is subject to potential anomalies — so too is the very definition of
the composite Q operator itself. We shall investigate these anomalies in section 3.3 below.
BRST invariant configurations may be found by setting the fermionic fields to zero
and asking that they remain zero under a BRST transformation. In this regard, the most
dangerous looking transformation is
δρIa =1
2νaZ
I (3.29)
which potentially forces Z to vanish, ruining the interpretation of our model as a map to
projective twistor space. Even if only some components of the supertwistor ZI were forced
to vanish, this would still place intolerable restrictions on the map and destroy any chance
of the model describing (non self-dual) gravity. Of course, the resolution is that in fact
ν = 0. To see that this is so, notice that since degL ≥ 0, the field νa ∈ Ω0(Σ,K1/2Σ ⊗L−1)
has no zero-modes (at least generically, and always at g = 0). Thus the path integral∫Dµ exp
(1
2π
∫Σµa∂ν
a
)(3.30)
over the non-zero modes of the conjugate fields µ imposes the constraint ν = 0, render-
ing (3.29) harmless. Likewise, the transformation δm ∼ 〈Z, Y 〉 + · · · is tame because Y
vanishes on-shell.
To summarize, our model contains just three field multiplets
Z ∈ Ω0(X,C4|N ⊗ L)
C ∈ ΠΩ0(X,D)
B ∈ ΠΩ0(X,Ber(X)⊗D∨) ,
(3.31)
and is defined by the action12
S =1
2π
∫X
d1|2z
(1
2〈Z, ∂Z〉+Ba∂C
a
)(3.32)
12Recall from (2.12) that Ber(X) ∼= L2 and that d1|2z is a holomorphic section of the trivial bundle
Ber(X)⊗ L−2 ∼= O.
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and the BRST charge Q of (3.24). This is enough to describe perturbative N = 8 super-
gravity, also allowing for gaugings and conformally flat backgrounds such as AdS4. The
only structures involved are the choice of infinity twistor 〈 , 〉 and the structure of X as
a complex supermanifold. As promised in section 2.1, zero modes of C represent (parity
reversed) global automorphisms of X that act trivially on Σ. Similarly, as in section 2.2,
parity reversed zero modes of B are Serre dual to H1(X,D), the tangent space to the
moduli space of X as a bundle over Σ.
3.3 Worldsheet anomaly cancellation
The worldsheet theory we have defined is chiral — the matter and ghost kinetic terms each
involve only the worldsheet Dolbeault ∂ operator — so it is potentially rife with anomalies.
We now investigate these, showing that all (local) anomalies cancel when N = 8.
We first compute the anomalies in the worldsheet gauge theory. Consider first the
GL(1;C) transformations associated to the non-trivial line bundle L. On the two dimen-
sional worldsheet, this anomaly is governed by a bubble diagram with the charged chiral
fields running around the loop. It is thus determined by the sums of the squares of the
charges of the fields, weighted by a sign for fermions. The GL(1) charged fields are the Y Z
system, giving a contribution a = (4−N ) to the gauge anomaly, the two βγ systems each
giving a = 1, and the two µν systems that also contribute a = 1 each. Altogether we have
aGL(1) = (4−N ) + 2 + 2 = 8−N (3.33)
so the GL(1;C) gauge anomaly vanishes if and only if N = 8.
If the SL(2;C) bundle has non-trivial second Chern class, there is a further potential
gauge anomaly. From the matter fields, only the ρρ-system transforms non-trivially under
SL(2 C), in the antifundamental representation. The bosonic βγ- and µν-ghosts trans-
form in the fundamental (or antifundamental), while the fermionic mabnab-system is in the
adjoint. The anomaly coefficient is thus
aSL(2) = −1
2(4−N ) trF(tktk) + 2 trF(tktk)− tradj(t
ktk) , (3.34)
where tk denote the generators of SL(2;C), in the representation indicated by the subscript
on the trace. (A sum over k is implied.) Note that the ρρ contribution has a symmetry
factor 12 since they are their own antiparticles. Writing trR(tktk) = C2(R) dim(R) in terms
of the quadratic Casimir of the representation, (3.34) becomes
aSL(2) =N2× 2C2(F)− 3C2(adj) =
3
4(N − 8) , (3.35)
where we used the SL(2;C) quadratic Casimirs C2(F) = 34 and C2(adj) = 2. The worldsheet
GL(2;C) gauge theory is thus completely free from local anomalies when N = 8.
In addition, we can compute the total Virasoro central charge.13 From the matter
fields, the Y Z-system contributes central charge c = 2(4−N ), twice the (complex) bosonic
13Since our theory does not involve worldsheet gravity — although there is a worldsheet gravitino asso-
ciated to the βγ-system — the role of this anomaly is not completely clear to me. Its vanishing nonetheless
seems significant.
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dimension of the non-projective target space, minus twice the fermionic dimension. The
ρρ-system are spin 12 fields on the worldsheet of opposite statistics to Z. As in the RNS
superstring, they contribute a further (4 − N ) to the central charge. From the ghosts
we have two bosonic βγ-systems each contributing c = +11 as in the RNS string, while
fermionic ghosts for the gauge system contribute c = −2 × dim(GL(2)). Finally, because
the spin 12 µν-systems are bosonic, they contribute c = −1 each. The total central charge
is thus
c = 3(4−N ) + 22− 8− 2 = 3(8−N ) (3.36)
and vanishes if and only if N = 8. There is in addition a potential mixed GL(1;C)-
gravitational worldsheet anomaly b which equals 8 −N and again vanishes when N = 8.
Since trR(tk) = 0 for SL(2;C), there is never any mixed SL(2)-gravitational anomaly.
For an alternative (though equivalent) view of things, performing the path integral over
the non-zero modes of all fields leads to determinants of ∂-operators. These ∂-operators
act on sections of bundles as appropriate for the charges and spins of the fields, and the
determinants appear in the numerator or denominator according to whether the fields
are fermionic or bosonic. The resulting chiral determinants are not functions, but form
a section of a determinant line bundle over the moduli space of the gauge theory (and,
in principle, the complex structure of the worldsheet). In order to make sense of the
determinants as these moduli vary, we must find a flat connection on this determinant line
bundle. A natural connection was provided by Quillen [89]. Its curvature F is computed
by the Freed-Bismut formula [90, 91]
F =
∫Σ
Td(TΣ) ∧ Ch(E) , (3.37)
where the bundle E depends on the fields in question. Letting x denote the first Chern
class of TΣ, y denote c1(L), and G denote the SL(2;C) bundle, we find
F =c
24
∫Σx ∧ x+
b
2
∫Σx ∧ y +
a
2
∫Σy ∧ y − s
∫Σ
c2(G) (3.38)
where aGL(1), aSL(2), b, and c are the anomaly coefficients computed above.
These potential anomalies would also afflict the BRST charge Q in (3.25), since it
is a composite operator. For example, the terms proportional to the ghost field n each
contain potential short distance worldsheet singularities. However, the coefficient of this
short distance singularity in the combination n (〈Y,Z〉 − βaγa + µaνa) is the same anomaly
coefficient a = (4−N ) + 2 + 2 as before, so Q is well-defined when N = 8. Similarly, the
terms proportional to nab are sensitive to any anomaly in the SL(2;C) gauge symmetry.
Finally, there are short distance singularities that potentially obstruct Q2 = 0, so that Q
could not be used as a BRST operator. It is left as an exercise to show that these again
cancel when N = 8.
3.3.1 Zero modes
The absence of gauge anomalies means the path integral over the complete set of non-zero
modes of all fields is well-defined, providing a section of a determinant line bundle over the
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JHEP04(2020)047
moduli space whose Quillen connection is flat. We now examine the properties of the zero
modes of the fields.
Consider first the charged fields in our theory. For a generic chiral βγ-system, with γ
taking values in some vector bundle E, Serre duality and the Riemann-Roch theorem give
nγ0 − nβ0 = h0(Σ, E)− h1(Σ, E) =
∫Σ
c1(detE) +1
2c1(TΣ) . (3.39)
In the case at hand, writing d for the degree of L, we find
nZ0 − nY0 = 4|N × (d + 1− g) (3.40)
for the bosonic and fermionic components of the Y Z-system at genus g,
nγ0 − nβ0 = 2(d + 2− 2g) (3.41)
in total for the two βγ-systems in our theory, and a total of
nµ0 − nν0 = 2d (3.42)
for the two µν-systems.14
The first important consequence of this calculation is that the path integral measure
D(Z, Y, β, γ, µ, ν)0 over these zero modes has net charge
(4−N )(d + 1− g) + 2(d + 2− 2g) + 2d = (8−N )(d + 1− g) , (3.43)
where we recall that by Berezin integration, the integral form dθ scales oppositely to θ
for a fermion. Thus, when N = 8, the zero mode path integral measure provides a top
holomorphic form on the moduli space of the theory.
However, while the total charge of all the zero modes cancels, there are selection rules
associated to the zero modes of the individual fields. Generically, when d is sufficiently
larger than g (and always at g = 0) the Kodaira vanishing theorem asserts that the above
indices are entirely due to the positively charged fields. Let us examine the consequences
of these selection rules, concentrating on this generic case.
To begin with, we have (d+1−g) zero modes of ZA = χA for each A = 1, . . . ,N running
over the fermionic directions of twistor space. These fermionic zero modes cause the path
integral to vanish unless they are saturated by insertions from the vertex operators. Exactly
as in the original twistor string [12, 56], this leads to a relation between the degree of the
curve Z(Σ) ⊂ CP3|N and the allowed helicity sector represented by the vertex operator
insertions. We will find that the vertex operators describe an N = 8 gravity supermultiplet,
in conventions where the positive helicity graviton is at order (χ)0 and the negative helicity
graviton is at (χ)8. Thus we find the usual relation
d = k + 1 + g (3.44)
14Here it is the antighost µ that has zero modes, at least generically.
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JHEP04(2020)047
between the MHV level k (‘number of negative helicity gravitons, minus 2’) and the degree
of the curve and genus of the worldsheet.15
Now we consider the zero modes of the bosonic fields. We have d + 2− 2g zero modes
for each of the two components of γa and d zero modes of each of the two µas. Of course,
we also have (d + 1 − g) zero modes for each bosonic twistor component. In the absence
of insertions that depend on these zero modes, the integrals over γ and µ would diverge,
rendering the path integral ill-defined. As in the RNS string, we will find that the vertex
operators have δ-function support in these fields, giving a meaningful path integral. For
now, recall from the introduction that in the flat space limit, an n-particle g-loop amplitude
with n± gravitons of each helicity is a monomial of degree n−−1 +g in the infinity twistor
〈 , 〉 as a form, and n+ − 1 + g in the infinity twistor [ , ] as a Poisson structure.16 With
the aid of (3.44), we may rewrite these numbers as
n− − 1− g = d
n+ − 1− g = n− (d + 2− 2g) ,(3.45)
respectively coinciding with the number of zero modes of each component of µ, and n minus
the number of zero modes of each γ component.
The remaining fields are the worldsheet spinors ρ, which generically have no zero
modes, and the ghost system associated with gauging the GL(2;C) transformations. These
ghosts certainly do have zero modes. However, at g = 0 we will content ourselves to treat
these by simply ‘dividing by vol(GL(2))’ — the path integral will lead yield a form that is
invariant and basic with respect to a natural GL(2) action, and we descend to the moduli
space. While this suffices to recover the g = 0 scattering amplitudes of [32], it is really too
naive. We discuss this further in section 5.
3.4 Vertex operators
In this section we construct the vertex operators representing BRST cohomology classes.
These will correspond to a linearized N = 8 supergravity multiplet. We also construct
picture changing operators required to fix zero modes of the bosonic antighosts.
3.4.1 The N = 8 supergravity multiplet
The odd supervector field V of (2.14) generates a global holomorphic automorphism of X
when V ∈ H0(X,D). In the generic case that d g, only the lowest two components v
and R in the superfield expansion of V can be globally holomorphic, with
v ∈ H0(Σ,D) and R ∈ H0(Σ,End(C2 ⊗ L)) . (3.46)
The fermionic symmetries of X→ Σ correspond to zero modes of the ghosts γ, with each γa
being one of the d+2−2g holomorphic sections of K−1/2Σ ⊗L, while the bosonic symmetries
15The genus h of the image curve Z(Σ) obeys h ≤ g. In particular at MHV level for g = 1, the worldsheet
must double cover a twistor line, branched over four points.16This monomial behaviour of course refers to the amplitude when written in twistor space. It becomes
obscure on momentum space because the transformation from twistors to momenta itself involves the infinity
twistor. Recall also that I is degenerate in the flat space limit.
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correspond to the zero modes of the ghost Nab that are constant. To obtain a moduli space
whose (virtual) dimension is non-negative, and hence to have a well-defined ghost path
integral, we must fix these zero modes.
In our case, the odd vector field v = va∂/∂θa s generates (smooth) translations of along
the fibres of X→ Σ. To fix the associated odd automorphisms of X we pick points pi ∈ Σ
and demand that the translations act trivially at these points. In the path integral, these
translations are represented by the ghosts γa, so we can force the translation to be trivial
at some pi by inserting17
δ2(γ) = δ(γ1) δ(γ2) (3.47)
at this point. Each such constraint reduces the dimension of the space of automorphisms
by 1, so generically we need to pick (at least) d+2−2g points to remove all the global odd
automorphisms. The resulting δ-functions absorb the γ zero modes, rendering the path
integral meaningful.
In the usual case of the RNS superstring, the vertex operator would also include a
factor of the fermionic c ghost instructing us to quotient the path integral only by those
diffeomorphisms of the bosonic Riemann surface Σ that act trivially at the pi. For us
however, since we only quotienting by diffeomorphisms generated by sections of D ⊂ TX,
there are no c ghosts. If we do not wish our answer to depend on the choice of pi ∈ Σ, we
must integrate over them.18 Because γa has spin −12 and charge +1 under L, the operator
δ2(γ) should be interpreted as a (1,0)-form of charge −2. So the simplest type of vertex
operators are
Oh ≡∫
Σδ2(γ)h(Z) , (3.48)
where h(Z) is a (0,1)-form on Σ of charge +2. This integral is to be taken over Σ at
θa = 0. These vertex operators are closely analogous to Neveu-Schwarz vertex operators
in the superstring.
The field h is a twistor representative of an N = 8 supergravity multiplet, pulled back
to Σ. Writing ZI = (Za|χA) for the bosonic and fermionic components of the twistor, we
can expand h as
h(Z|χ) = h(Z) + χAψA(Z) +1
2χAχBaAB(Z) + · · ·+ (χ)8 h(Z) (3.49)
where the coefficient of (χ)p is a (0, 1)-form on twistor space of homogeneity 2−p. Via the
linearized Penrose transform [92–94], these states correspond to massless fields with one
boson of helicity +2, 8 fermions of helicity + 32 , 28 gauge fields of helicity +1 and so on
until we reach the field h that corresponds to a negative helicity graviton. More precisely,
the Penrose transform asserts that an on-shell, linearized N = 8 supergravity multiplet
corresponds to a cohomology class in H1(PT,O(2)), of which h is a representative.
17The fact that our vertex operators can depend only on γ, and not on derivatives of γ, is determined
by the requirement that the path integral measure over all fields does actually descend to a measure — or
top holomorphic form — on the moduli space of X→ D. See [55] for an explanation in the context of RNS
superstrings.18See however the discussion in section 5.1.
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While Oh is to be inserted on a fixed section of X → Σ (taken to be the zero section
θa = 0), we also have related vertex operators Oh that are integrated over the entire
worldsheet supermanifold X. As usual, these are obtained simply by replacing the factor
of δ2(γ) in (3.48) by the integration measure d1|2z on X. We have
Oh ≡∫
Xd1|2z h(Z)
=
∫Σ
∂h(Z)
∂ZIY I − 1
2εbaρIaρ
Jb
∂2h(Z)
∂ZI∂ZJ.
(3.50)
Note that if I, J correspond to bosonic twistor directions, then ρI and ρJ anticommute, and
the εab symbol ensures that the second term here is symmetric in I, J . Conversely, if I, J
are fermionic directions, the expression is naturally antisymmetric in I, J . In particular,
this means that (3.50) is well-defined as a composite operator, with no short distance
singularities — the potential singularity in either the ρρ-system or the Y Z-system are each
proportional to the (graded) antisymmetric infinity twistor I, so the resulting derivatives
on h would vanish.
The first term in Oh also has a natural meaning in twistor space: since h represents
an element of H1(PT,O(2)), and in canonical quantization of the action (3.11) we have
Y ∼ ∂/∂Z, the term ∂h∂ZI
Y I represents an element of H1(PT, TPT). It therefore describes
an infinitesimal deformation of the complex structure of twistor space. The non-linear
Penrose transform [81] asserts that performing a finite deformation of the complex struc-
ture of twistor space corresponds to turning on self-dual Weyl curvature in space-time.
The holomorphic geometry of twistor space determines the conformal structure of space-
time, so an arbitrary deformation of this complex structure, generated by an arbitrary
V ∈ H1(PT, TPT), corresponds to an arbitrary self-dual solution of the Bach equations of
conformal gravity. However, unlike in the original twistor string, the vertex operators we
have found here are associated not to arbitrary vector fields, but rather to vector fields19
V Ih ≡ IIJ
∂h∂ZJ
that are Hamiltonian20 with respect to the Poisson structure defined by I.
Deforming the complex structure by a such Hamiltonian vector field ensures that the holo-
morphic Poisson structure is preserved, and hence the corresponding deformed space-time
still has a preferred metric. This metric is a self-dual solution of the vacuum Einstein
equations [81–83]. Extending this to the N = 8 multiplet gives a BPS solution to the field
equations of supergravity.
As usual, the integrated vertex operator (3.50) can be added to the original action
S1 → S′1 =
∫X
d1|2z(〈Z, ∂Z〉+ h(Z)
)=
∫Σ〈Y, (∂Z + Vh)〉 + fermions ,
(3.51)
describing strings propagating on a background twistor space with deformed complex struc-
ture ∂ → ∂ + Vh. In the twistor string framework, deformations that are not self-dual are
described perturbatively in terms of higher degree maps (worldsheet instantons).
19We have used I to put the vector field index in the natural place. This is harmless when I is non-
degenerate, and in section 4.1 we shall see it happens automatically in the flat space limit.20This is the reason we denote the gravity multiplet by h.
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JHEP04(2020)047
To summarize, freezing all the γ zero modes requires us (in the generic case) to include
d + 2 − 2g vertex operators of the form Oh, involving the basic twistor wavefunction h.
The remaining n− (d + 2− 2g) states are represented by integrated vertex operators Ohthat involve not h itself, but rather its associated Hamiltonian vector field Vh. In the
introduction we saw that the flat space tree-level scattering amplitude involves precisely
n− (d+ 2− 2g) powers of the infinity twistor [ , ] (i.e., the infinity twistor as a degenerate
Poisson structure). In particular, at g = 0 the tree amplitudes (1.5) are monomials of
degree n − d − 2 in [ , ]. We now understand that this fact has its origin in the odd
automorphisms of X as the total space of a fermionic bundle over a fixed Σ.
3.4.2 Picture changing operators
The worldsheet supermanifold X also has moduli, even for a fixed Riemann surface Σ,
coming from the freedom to deform the distribution D ⊂ TX. As in section 2.2, for fixed Σ
the tangent space to this moduli space is H1(X,D). Our twistor string knows about these
moduli via the zero modes of the antighost multiplet B, which live in the parity reversed
Serre dual group ΠH0(X,Ber(X)⊗D∨) ∼= H0(X,C2⊗K1/2Σ ⊗L). In the generic case with
d g, the only components of the B multiplet to have zero modes are µ and M.
Consider first the odd moduli space, associated to zero modes of the bosonic µ antighost
in H0(Σ,C2 ⊗ K1/2Σ ⊗ L). To integrate over the odd moduli space we follow the usual
procedure of the RNS superstring and insert 2h0(Σ,K1/2Σ ⊗L) ‘picture changing operators’
Υ ≡ 2 δ2(µ)SaSa , (3.52)
where
Sa ≡1
2〈Z, ρa〉+
1
2nµa − n b
a µb + mγa + mabγb (3.53)
is the supercurrent obtained by taking the BRST transformation of µa as in (3.28). These
insertions correspond to δ-function wavefunctions for the gauginos associated to the µν-
system, and fix the µa zero modes. See e.g. section 3 of [55] for a clear discussion of picture
changing operators and their relation to fixing parity odd moduli.
Unlike the usual picture changing operators of the RNS string, the operator (3.52)
involves two copies of these currents because D has rank two. Because x δ(x) = 0 we can
neglect the terms proportional to the µ antighost in these supercurrents. Similarly, at
g = 0 when the antighosts Mab have no zero modes (and there are no N insertions with
which to contract), we can neglect their contribution to Sa. Then the picture changing
operator simplifies to become
Υ ≡ 1
2δ2(µ) 〈Z, ρa〉 〈Z, ρa〉 (3.54)
When g = 0 we need (a minimum of) d such Υ insertions.
The previous expression may be thought to be somewhat formal, because composite
operator 〈Z, ρ1〉〈Z, ρ2〉 has a potential short distance singularity from the ρρ contraction.
To regularize this, we point split the two 〈Z, ρ〉 factors and take a limit as they come
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JHEP04(2020)047
together. In terms of a local coordinate z on U ⊂ Σ, one finds
limz′→z
⟨〈Z, ρ1(z′)〉 〈Z, ρ2(z)〉
⟩ρρ
= limz′→z
√dz′√
dz
z′ − z× 〈Z(z′), Z(z)〉 , (3.55)
where the factors of√
dz′ and√
dz arise since ρ is a spinor on Σ. Expanding the holomor-
phic field Z as ZI(z′) = ZI(z) + (z′ − z) ∂zZI(z) + · · · , we see that the pole from the ρρ
propagator is cancelled by a zero from the antisymmetric infinity twistor, leaving us with
a finite contribution
limz′→z
⟨〈Z, ρ1(z′)〉 〈Z, ρ2(z)〉
⟩ρρ
= −〈Z, dZ〉(z) . (3.56)
We now define Υ more precisely as the normal ordered operator
Υ ≡ δ2(µ)
(1
2:〈Z, ρa〉〈Z, ρa〉: − 〈Z, dZ〉
)(3.57)
in which the local contribution of the ρρ-system is explicitly accounted for. The normal
ordering prescription : : is understood to mean that we do not consider contractions
between the enclosed fields.
Actually, since the potential short distance singularity cancelled in (3.55), we are free
to think of Υ as in (3.54) without normal ordering. We must then remember to include the
local contribution (3.56) when computing correlation functions involving these operators.
In practice, this approach turns out to be somewhat simpler.
Of course, we could have chosen to represent all the external states by the vertex
operators Oh of (3.48), rather than use any integrated ones Oh. In this description, we
would quotient the path integral only by translations of Σ inside X that act trivially at
n > d + 2 − 2g points. Since we are quotienting by fewer fermionic symmetries, the odd
dimension of the moduli space increases and we must integrate over this larger odd moduli
space. In the language of ghosts, the additional insertions of Oh provide extra factors of
δ2(γ). To compensate for these we should also construct picture changing operators for
the βγ-system, inserting n− (d + 2− 2g) of them so as to provide a top form on the odd
moduli space. This is expected to be the correct approach if one wishes to obtain a detailed
understanding of the compactification of this moduli space [55, 78]. It would be interesting
to investigate this further.
Finally, when g ≥ 1 we also have zero modes of the fermionic antighosts Mab. Insertions
of these amount to constructing a top holomorphic form on the (bosonic) moduli space of
holomorphic GL(2;C) bundles on Σ. For the (generic) case that this bundle is stable and
g ≥ 2, this moduli space has dimension 3(g− 1) + g and has been extensively studied [95–
102]. We discuss it further in section 5.1.
4 Scattering amplitudes in the flat space limit
Our prescription for computing n-point worldsheet correlation functions in the g = 0
twistor string is to consider the path integral⟨d+2∏j=1
Ohjn∏
k=d+3
Ohkd∏l=1
Υl
⟩=
⟨d+2∏j=1
∫Σδ2(γ)hj(Z)
n∏k=d+3
∫X
d1|2z hk(Z)
d∏l=1
Υl
⟩.
(4.1)
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In this section, we will use this prescription to recover the flat space tree-level S-matrix
of N = 8 supergravity in the form obtained in [32]. To do so, we will need to be able
to handle correlation functions of βγ-systems involving operators such as δ2(γ). A clear
explanation of how to achieve this was recently provided in [55] (see especially section 10).
For convenience, the relevant points are summarized in appendix A.
4.1 A degenerate infinity twistor
To compute scattering amplitudes, we must take the limit as the cosmological constant Λ
tends to zero. In this limit, the rank of the infinity twistor
IIJ =
Λεαβ 0 0
0 εαβ 0
0 0√
ΛδAB
(4.2)
we have been working with so far becomes non-maximal. In particular, in the flat space
limit we must carefully distinguish between the infinity twistor as a form and the infinity
twistor in its role as a bivector, since
IIJIJK = Λ δ KI → 0 (4.3)
and so they are not equivalent.
If we were to take the flat space limit naively, the matter action (3.11) would also be-
come degenerate, with the kinetic terms for the µα and χA components of the supertwistor
ZI vanishing. To avoid this, before taking the limit, we relabel the fields as
ZI → ZI ρI1 → ρI ρI2 → IIJ ρJ Y I → IIJYJ (4.4)
and include an overall factor of 1/Λ in the normalization of (3.11). In terms of the rescaled
fields, the matter action becomes
S1 =1
2π
∫ΣYI∂Z
I + ρI∂ρI (4.5)
and is independent of the cosmological constant. The ghost fields are unchanged. Having
ensured the action remains non-degenerate, we can now freely take Λ → 0, setting21
IIJ =
εαβ 0 0
0 0 0
0 0 0
and IIJ =
0 0 0
0 εαβ 0
0 0 0
. (4.6)
We follow the standard convention that 〈 , 〉 denotes contraction by IIJ with downstairs
indices, involving only the λ part of Z, whereas [ , ] denotes contraction by IIJ with
upstairs indices and involves only the derivatives ∂/∂µ tangent to twistor space (or the λs
on momentum space).
21In the presence of an arbitrary gauge coupling g, we are taking the limit Λ → 0, g√
Λ → 0. See the
discussion at the end of section 3.1.1.
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While the rescaled action is independent of the infinity twistor, the same cannot be
said for the BRST operator
Qmatter =1
2
∮d1|2z 〈Z, DaZ〉 . (4.7)
As with the action, we first apply the recaling (4.4) with a non-degenerate infinity twistor
and then take the limit Λ→ 0. The matter BRST charge becomes
Qflat =
∮γ1YIρ
I + γ2[Y, ρ] +1
2ν1〈ρ, Z〉+
1
2ν2ρIZ
I
+1
2nYIZ
I +1
2
(n12 + n21
)ρI ρI +
1
2n11〈ρ, ρ〉+
1
2n22[ρ, ρ] ,
(4.8)
and the presence of the degenerate infinity twistor means that not all supertwistor com-
ponents appear in all terms; for example, 〈ρ, Z〉 = ραλα while [Y, ρ] = Yαρα. A somewhat
similar BRST operator occurs in Poisson sigma models, see e.g. [52, 53]. It would be in-
teresting to explore this connection further. Again, the ghost BRST charge is unaltered.
Similarly, in the flat space limit the vertex operators become
Oh =
∫Σδ2(γ)h(Z)
Oh =
∫Σ
[Y,∂h
∂Z
]+
[ρ,
∂
∂Z
(ρI
∂h
∂ZI
)] (4.9)
for the external states and
Υ = δ2(µ) 〈ρ, Z〉 ρIZI (4.10)
for the picture changing operator. As promised, the integrated vertex operator Oh naturally
depends on the Hamiltonian vector field[∂h∂Z ,
]associated to the infinity twistor as a
Poisson structure.
4.2 The worldsheet Hodges matrix
We are now in position to recover the flat space gravitational S-matrix from the corre-
lator (4.1) at g = 0. In this section we will show that the worldsheet Hodges matrix Hin (1.6) arises from the correlation function of the matter vertex operators Oh and Oh.
Firstly, we notice that the only insertions of γ come from the δ-functions in the fixed
section vertex operators Oh. These δ-functions serve to fix the integrals over the zero
modes of γa, representing elements of H0(Σ,K−1/2Σ ⊗ L). For each flavour γ1 and γ2, we
expand γ as
γa(σ) =
d+2∑i=1
Γa iYi(σ) + non-zero modes , (4.11)
where the Yi form a basis of the zero modes (written in terms of a homogeneous coordinate
σα on the CP1 worldsheet), and where the Γas are c-number constants. In [55], it was
explained that for each flavour of γa, the insertion of δ-functions leads to⟨d+2∏j=1
δ(γ(σj))
⟩βγ
=1
det(Y). (4.12)
– 27 –
JHEP04(2020)047
See also the discussion in the appendix. Here, Y is the (d + 2) × (d + 2) matrix whose
entries are Yij = Yi(σj). At genus zero, a basis of H0(Σ,K−1/2Σ ⊗ O(d)) is given by
Yi(σ) = (σdσ)−1/2σα1 · · ·σαd+1 . Computing this determinant and including both flavours,
the path integral over the βγ-system yields⟨d+2∏j=1
δ2(γ(σj))
⟩βγ
=1
|σ1 · · ·σd+2|2×
d+2∏j=1
(σjdσj) , (4.13)
where |σ1 · · ·σd+2| denotes the Vandermonde determinant
|σ1 · · ·σd+2| ≡∏i<j
i,j∈1,...,d+2
(ij) . (4.14)
This Vandermonde determinant is precisely the denominator factor (1.7) of det′(H) in the
introduction, here specialized to the case that we remove the first d + 2 rows and also the
last d + 2 columns in computing a minor of the Hodges matrix H. (That is, we compute
the (d + 3)rd principal minor.)
The (n − d − 2) × (n − d − 2) minor of H itself comes from the remaining part of
the matter vertex operators. As a first step to understanding this, consider the n− d− 2
insertions of Oh and temporarily neglect the[Y, ∂h∂Z
]terms. The remaining part of Oh is
bilinear in the worldsheet spinors ρ and ρ. These are free fields on Σ. Since they have no
zero modes, all ρ and ρ insertions must be absorbed by contracting them pairwise in all
possible combinations. Insertions of ρ and ρ can be found both in Oh and in the picture
changing operators Υ. However, with our degenerate infinity twistor, Oh involves only the
α components of ρ, while Υ involves only the α components of ρ. The off-diagonal two
point function 〈ρα(σ) ρα(σ′)〉 vanishes, so the ραs from any Oh insertion can contract only
with the ραs present in some other Oh insertion. Furthermore, the pieces ρα∂h∂λα
+ρA ∂h∂χA
in
ρI ∂h∂ZI
in (4.9) may be ignored because there is always at least one unpaired ρα left over in
some Oh and at least one unpaired ρα left over in some Υ, either of which causes the path
integral to vanish. We conclude that we can consider the ρρ factors in Oh independently
from those in Υ, and that we only need consider the α terms in Oh.
This being so, consider the correlator22
C(σd+3, . . . , σn) ≡
⟨n∏
k=d+3
[ρ,
∂
∂Z
(ρβ∂hk
∂µβ(σk)
)]⟩ραρα
=
⟨n∏
k=d+3
ραρβ∂2hk
∂µα∂µβ(σk)
⟩ραρα
(4.15)
coming purely from the ρρ-system. In terms of the homogeneous worldsheet coordinates
σα, the two-point function of the ρρ-system is
〈ρα(σi) ρβ(σj)〉 = δαβ
(σidσi)12 (σjdσj)
12
(ij). (4.16)
22In this expression µ denotes the bosonic twistor component, not the antighost, as should be clear from
the indices.
– 28 –
JHEP04(2020)047
Using this propagator to perform all possible contractions in (4.15) yields
C(σd+3, . . . , σn) =∣∣∣H(0)
(n−d−2)×(n−d−2)
∣∣∣ n∏k=d+3
hk(Z(σk)) (σkdσk) , (4.17)
where∣∣∣H(0)
(n−d−2)×(n−d−2)
∣∣∣ is the (d + 3)rd principal minor of the matrix H(0) whose ele-
ments are
H(0)ij =
1
(ij)
[∂
∂Zi,∂
∂Zj
]=
1
(ij)
[∂
∂µi,∂
∂µj
]for i 6= j (4.18)
and zero on the diagonal. In (4.18) we understand that ∂/∂Zi acts on hi in the prod-
uct (4.17), differentiating this wavefunction with respect to (the µα component of) its
argument Z(σi). For example, if the external states are taken to be twistor representatives
hi(Z(σi)) =
∫dsis3i
δ2(λi − siλ(σi)) exp
(siµ
α(σi)λiα + siχA(σi)ηiA
)(4.19)
of momentum eigenstates, then H(0)ij → sisj [ij]/(ij).
H(0) is not quite the full worldsheet Hodges matrix (1.6), because the diagonal elements
H(0)ii vanish. To recover (1.6) in its entirety, we now consider the additional effect of the[Y, ∂h∂Z
]terms in the integrated vertex operators (4.9). With the degenerate infinity twistor,
only the α component of Y is present here. This Yα cannot contract with any Z in Υ,
either because there is no short distance singularity or because we would again be left
with an unpaired ρ. However, the Yα from any given Oh may contract with the Zs in
the wavefunctions h(Z) in any of the remaining Oh operators, and also in the ‘fixed’ Ohvertex operators.
Since Y ∈ Ω0(Σ,KΣ ⊗O(−d)) and Z ∈ Ω0(Σ,O(d)), the Y Z-propagator is
〈YI(σi)ZJ(σj)〉 = δ JI
(σidσi)
(ij)
d+1∏r=1
(arj)
(ari), (4.20)
where the σar in the product on the right are arbitrary. This product ensures that both
sides have the correct homogeneity. It arises because ∂−1f is ill-defined if f is a (0,1)-form
of homogeneity d > −1; we are always free to modify ∂−1f → ∂
−1f + g where g is an
arbitrary holomorphic section of O(d), since this is annihilated by ∂. We can fix a choice
of propagator by specifying d + 1 points at which ∂−1f vanishes. This is the role of the
product in (4.20). Of course, any meaningful expression — such as the Hodges matrix —
is independent of the choice of these points (see [26, 32, 33]).
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JHEP04(2020)047
Using this propagator, with just one[Y, ∂h∂Z
]insertion we have⟨[
Y,∂hd+3
∂Z(σd+3)
] n∏j=d+4
ραρβ∂2hj
∂µα∂µβ(σj)
d+2∏i=1
hi(Z(σj))
⟩
=∣∣∣H(0)
∣∣∣ ⟨[Y, ∂hd+3
∂Z(σd+3)
] d+2∏i=1
hi(Z(σi))
⟩n∏
j=d+4
(σjdσj)
=∣∣∣H(0)
∣∣∣−
n∑k=1
k 6=d+3
1
(d+3 k)
[∂
∂µd+3,∂
∂µk
] d+1∏r=1
(ar k)
(ar d+3)
×
n∏i=1
hi(Z(σi))
n∏j=d+3
(σjdσj)
(4.21)
where in the first step we integrated out the ρ fields using (4.17). The term in braces in
the last line is one of the diagonal elements (the (d + 3)rd diagonal entry) that we were
missing from the full Hodges matrix.
We now prove inductively that the sum of Y and ρρ insertions in each integrated
vertex operator Oh means that the total correlation function assembles itself into the
complete worldsheet Hodges matrix H. To start, this is certainly true when there are only
2 integrated vertex operators (corresponding to an n-point MHV amplitude). For⟨2∏i=1
([Y,∂hi∂Z
]+ ραρβ
∂2hi
∂µαi ∂µβi
)n∏j=3
hj
⟩
=
(H11H22 +
∣∣∣∣∣ 0 H12
H21 0
∣∣∣∣∣)
(σ1dσ1)(σ2dσ2)
n∏i=1
hi
=
∣∣∣∣∣H11 H12
H21 H22
∣∣∣∣∣ (σ1dσ1)(σ2dσ2)
n∏i=1
hi ,
(4.22)
where the first equality follows because we must either take the ρρ term at both sites or at
none23 (If there is only one Oh insertion — which occurs only for the 3-pt MHV amplitude
— we are forced to take the Y contribution as in (4.21); the ρ fields cannot contribute
at all.)
To perform the induction, assume that the worldsheet correlator correctly gives the
determinant of the full Hodges matrix when there are m− 1 Oh operators, for some value
of m. Then from (4.21)⟨[Y,∂h1
∂Z1
] m∏j=2
Ohjn∏
k=m+1
hk
⟩= H11 ×
∣∣H(m−1)×(m−1)
∣∣× n∏i=1
hi
m∏j=1
(σjdσj) , (4.23)
23To lighten the notation, we have temporarily reversed our convention and taken the first n − d − 2
vertex operators to be integrated and the last d+ 2 to be fixed. This corresponds to computing the leading
principal minor of H.
– 30 –
JHEP04(2020)047
where the sum in each of the diagonal elements of the (m−1)×(m−1) minor∣∣H(m−1)×(m−1)
∣∣also runs over site 1, since the Y insertions in the Ohj s leading to this matrix may addi-
tionally contract with site 1. The Hodges factors on the right hand side of this expression
can be written as the determinant of an m×m symmetric matrix with H1j = 0 for j 6= 1.
On the other hand, instead choosing the ρρ term at site 1 gives24
⟨ραρβ
∂2h1
∂Zα1 ∂Zβ1
m∏j=2
Ohjn∏
k=m+1
hk
⟩= det
0 H12 H13 · · · H1m
H12 H22 H23 · · · H2m
H13 H23. . .
......
...
H1m H2m · · · Hmm
n∏i=1
hi
m∏j=1
(σjdσj)
(4.24)
where the first row and first column represent the possible choices of contraction for the
additional ρ and ρ insertions. As before, the sum in the diagonal entries of the (m− 1)×(m − 1) Hodges matrix should be extended to run over site 1. It is now clear that (4.23)
& (4.24) combine to give the determinant of the full Hodges matrix appropriate to m
insertions of Oh and n−m insertions of Oh.
Combining this with the factor (4.13) from fixing the γ zero modes shows that the
vertex operators (4.9) contribute⟨d+2∏i=1
Ohin∏
j=d+3
Ohj
⟩=
∫det′(H)
n∏i=1
hi(Z(σi)) (σidσi) (4.25)
to the twistor string path integral, in the specific case that we choose to remove the same
d+ 2 rows and columns in computing a minor of the full n×n Hodges matrix (here chosen
to be rows and columns 1 through d + 2). In [26, 32] we are actually free to compute
any (n − d − 2) × (n − d − 2) minor of H, provided we divide by the corresponding two
Vandermonde determinants. We can arrive at these more general representations by also
allowing ‘intermediate’ vertex operators that involve a single δ(γ) and an integral over the
other θ. That is, if we wish to compute a minor of the Hodges matrix involving different
rows and columns, we should allow∫dθ2 δ(γ1(σ)) h(Z)|θ1=0 and
∫dθ1 δ(γ2(σ)) h(Z)|θ2=0 (4.26)
as well as Oh and Oh. The rows and columns that we remove from the Hodges matrix
correspond to the independent insertion points of δ(γ1) and δ(γ2). More generally, it should
be clear that since the amplitudes depend on H only through det′(H), there is actually a
very large amount of freedom in the elements themselves. It would be interesting to know
if these examples considered in [103] can be realized on the worldsheet. Of course, since
det′(H) is invariant under arbitrary permutations of all n external states, the minimal case
considered in (4.25) is sufficient to recover the amplitude. Indeed, permutation invariance
24The lines in the matrix in (4.24) are simply to distinguish contributions from the new insertions at site
1 from the previous inductive step. H is not a supermatrix.
– 31 –
JHEP04(2020)047
is now seen to be a consequence of the usual fact that it does not matter which vertex
operators we choose to be ‘fixed’ and which ‘integrated’. Thus we have recovered one of
the main ingredients in the formula (1.5) for the tree-level gravitational S-matrix.
4.2.1 Self-dual N = 8 supergravity
In the following section we will show that the remaining, conjugate Hodges matrix comes
from the d insertions of picture changing operators Υ. However, in the special case that
d = 0 — corresponding to constant maps to twistor space — X has no odd moduli and no
Υs need be inserted. This case is worth investigating separately.
Since Z(σ) = Z for constant maps, we obtain the 3-point MHV amplitude⟨∫Σδ2(γ)h1(Z)
∫Σδ2(γ)h2(Z)
∫X
d1|2z h3(Z)
⟩=
∫D3|8Z ∧ h1(Z) ∧ h2(Z), h3(Z) ,
(4.27)
where we divided by vol(GL(2;C)) in lieu of fixing the zero associated to the worldsheet
gauge theory. This absorbs the integration over the three vertex operators over Σ and
ensures the remaining integral is taken over the projective twistor space. As usual, the
braces , denote the Poisson bracket associated to the infinity twistor I as a Poisson
structure IIJ ∂∂ZI∧ ∂
∂ZJ. Note that the Poisson bracket itself has homogeneity −2, while
each hi(Z) has homogeneity +2, so (4.27) is well-defined on the projective twistor space.
This 3-point MHV amplitude is especially important because it is the vertex of the
action
Ssd =
∫D3|8Z ∧
(h ∧ ∂h+
2
3h ∧ h, h
), (4.28)
evaluated on on-shell states. Ssd is the twistor action for self-dual N = 8 supergravity
and was first obtained by Mason & Wolf in [85]. At the linearized level its equations of
motion say that h represents an element of H(0,1)(PT,O(2)), as we have used in our vertex
operators. At the nonlinear level, the equations of motion assert that the almost complex
structure determined by ∂ + h, is integrable. Once again, the fact that we deform the
complex structure only by Hamiltonian vector fields ensures that we have a solution of
self-dual Einstein gravity, rather than self-dual conformal gravity [81, 82]. Ssd is clearly
analogous to the holomorphic Chern-Simons theory
SsdYM =
∫D3|4Z ∧ tr
(A ∧ ∂A+
2
3A ∧ [A,A]
)(4.29)
that describes self-dual N = 4 super Yang-Mills in twistor space [12]. Just as (4.29) is the
string field theory of the perturbative open B-model, we can interpret (4.28) as the string
field theory of our twistor string, restricted to constant maps.
4.3 The conjugate Hodges matrix
When d > 0 we must also account for the picture changing operators Υ =
δ2(µ)〈ρ1, Z〉ρ2IZI . These will supply the conjugate Hodges matrix H∨.
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JHEP04(2020)047
In the recipe (4.1) there are no insertions of µ or ν except for those in the picture
changing operators, which just suffice to absorb the µ zero-modes. Recall that at g = 0 a µ
zero-mode is an element of H0(Σ,K1/2Σ ⊗O(d)). Thus, at g = 0 we can expand each µa as
µa(σ) = (σdσ)1/2 Maα1···αd−1σα1 · · ·σαd−1 + non-zero modes , (4.30)
where the Ms are constants. Following the discussion of the appendix, we find from (A.13)
that the δ-functions associated to freezing the odd moduli produce a factor⟨d∏l=1
δ2(µ(σl))
⟩µν
=1∏d
l=1(σldσl)
1
|σ1 · · ·σd|2. (4.31)
The final factor is the Vandermonde determinant (1.9) appearing in the denominator of
det′(H∨) in (1.5), specialized to the case that we compute the first d × d minor of H∨.
Notice that when d = 1 the unique µ zero mode is µ(σ) = (σdσ)1/2 for each of the two
antighosts µ1 and µ2. Therefore, in the case of the MHV tree, the Vandermonde factor
in (4.31) is replaced by unity. This was the prescription taken in [32].
The numerator of the conjugate Hodges matrix comes from the associated supercur-
rents. These are 〈ρ, λ〉 ρIZI = 〈ρ, λ〉(ραµα + ραλα + ρAχA). Again, the ρρ-system has no
zero modes, so we must absorb all these insertions by contractions. Since 〈ρ, λ〉 cannot
contract with the α or A components of ρ, the only term in the bracket that can contribute
is ραλα.25 Using the propagator
〈ρα(σl)ρβ(σm)〉 = δ β
α
(σldσl)12 (σmdσm)
12
(lm)(4.32)
as in (4.16), performing all possible ρ–ρ contractions yields⟨d∏l=1
〈ρ, λ〉 ραλα(σl)
⟩=∣∣H∨d×d
∣∣× d∏l=1
(σldσl) , (4.33)
where∣∣H∨d×d
∣∣ is the first d× d minor of the matrix with elements
H∨lm =〈λ(σl), λ(σm)〉
(lm)for l 6= m, and H∨ll = −〈λ(σl), dλ(σl)〉
(σldσl). (4.34)
To obtain this result, recall that contractions of worldsheet fermions lead to a determinant
of a matrix whose l-mth entry corresponds to a propagator from site l to m. The diagonal
elements arise as in (3.56) since we are using the form of picture changing operator without
normal ordering, so must allow contractions between ρ and ρ at the same site — see the
discussion in section 3.4.2. Indeed, when d = 1 this is the only contribution.
The off-diagonal elements of H∨ in (4.34) are exactly the same as those in the conjugate
Hodges matrix (1.8). However, the diagonal elements in (1.8) and (4.34) appear to be
25Recall from section 4.2 that, with the degenerate infinity twistor of flat space, there could be no
contribution from cross-contractions of ρs in Υ with any ρ in Oh.
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JHEP04(2020)047
different. Let us now show that (1.8) can be simplified so that it takes the form (4.34).26
To begin with, notice that 〈λ(σl), λ(σm)〉/(lm) is everywhere finite, since the potential pole
is cancelled by a zero in the numerator. Now consider the diagonal term27
(σldσl)×∑m 6=l
〈λ(σl), λ(σm)〉(lm)
n−d−1∏r=1
(arm)
(arl)
∏k 6=l,m
(kl)
(km)(4.35)
of (1.8), where we have included a factor of (σldσl). The only possible poles in σl come
from the factors (arl) involving the reference points. However, a key point in [32, 33] was
that (4.35) was completely independent of these points (see [33] for a contour integral proof
of this). Therefore (4.35) is actually holomorphic in σl. Furthermore, (4.35) is a scalar of
homogeneity zero in all other points. Since the first factor in the sum is everywhere finite,
the only possible poles in σm come from the final product and so occur when pm collides with
some other marked point pk, with k,m 6= l. But for any given k,m (say m = 2 and k = 3),
it is easy to check that the singularity cancels in the sum. Therefore (4.35) has no poles in
any of the σm (and hence none in any of the σk). But by Liouville’s theorem, a function
homogeneous of degree zero that is everywhere holomorphic on a compact Riemann surface
must be constant. Quite remarkably, we have learnt that (4.35) is completely independent
of all the marked points except for σi. Finally, since (4.35) is both a (1, 0)-form in σl of
homogeneity 2d and is linear in the infinity twistor 〈 , 〉, we see that
−∑m 6=l
〈λ(σl), λ(σm)〉(lm)
n−d−1∏r=1
(arl)
(arm)
∏k 6=l,m
(kl)
(km)= −〈λ(σl), dλ(σl)〉
(σldσl). (4.36)
This is exactly H∨ll in (4.34).
With this simplification understood, combining (4.31) with (4.33) shows that the pic-
ture changing operators give⟨d∏l=1
Υ(pl)
⟩=
∣∣H∨d×d
∣∣|σ1 · · ·σd|2
= det′(H∨) (4.37)
as the factors of (σldσl) cancel. This is exactly the contribution of the conjugate worldsheet
Hodges matrix in (1.5), again represented by the specific case that we compute the first
d× d minor (the leading principal minor).
We now address an issue that may have been puzzling some readers. In the above,
we implicitly chose to insert the picture changing operators at d of the same points as the
matter vertex operators. Although this was the minimal choice, was it really necessary?
In fact, as in usual superstring theory, the picture changing operators may be inserted at
completely arbitrary points on the worldsheet, and these locations are not integrated over.
Rather than repeat the standard abstract argument for this (for which see [54, 104]), we
26I am greatly indebted to Lionel Mason for pointing this fact out to me, using a slightly different
argument to the one given here.27Recall that the sum here runs over all m ∈ 1, . . . ,n, m 6= l. Likewise, the final product is for all
k ∈ 1, . . . ,n except l and m.
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JHEP04(2020)047
shall show directly that despite appearances, (4.37) is in fact completely independent of
the choice of d insertion points.
As a warm-up, it is easy to see this claim is certainly true when d = 1, for then
det(H∨) = H∨11 = −〈λ(σ1), dλ(σ1)〉/(σ1dσ1) and the Vandermonde determinant is unity.
Since Z(σ) = Aσ0 + Bσ1 at MHV level, this becomes simply −〈A,B〉 which is obviously
independent of the insertion point.
For the general case, note first that (4.37) is homogeneous of degree zero in each of the
σls. The minor of H∨ itself can have no poles — it is a polynomial in its entries (4.34), each
of which are everywhere finite. Thus the only possible singularities in (4.37) come from
the Vandermonde determinant |σ1 · · ·σd|2 in the denominator. This produces a second
order pole when any pair of insertion points collide. We shall show that this singularity is
cancelled by a second order zero from∣∣H∨d×d
∣∣.To see this, suppose p1 approaches p2 with their separation measured by any small
parameter ε that has a first-order zero when they collide. Then for m ≥ 3 we have
H∨1m → H∨2m +O(ε). Subtracting rows and columns, the numerator of (4.37) becomes
∣∣H∨d×d
∣∣ =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
H∨11∗ H∨12
∗ O(ε) · · · O(ε)
H∨12∗ H∨22
〈2,3〉(23) · · · 〈2,d〉
(2d)
O(ε) 〈2,3〉(23)
. . .
......
...
O(ε) 〈2,d〉(2d) · · · H∨dd
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(4.38)
as ε becomes small. Here, H∨12∗ ≡ (H∨12 − H∨22). However, in section 3.4.2 H∨ii was defined
to be the limit of H∨ij as the two points collide, so H∨12∗ = O(ε) automatically. Similarly,
H∨11∗ ≡ H∨11 − 2H∨12 + H∨22 (4.39)
is by definition ε2 times the second derivative of H∨12 at σ1 = σ2, plus higher order cor-
rections. So H∨11∗ = O(ε2). Therefore, as p1 → p2 we can extract a factor of ε from the
first row and a separate factor of ε from the first column in (4.38), showing that the d× d
minor of H∨ indeed has a second order zero in this limit. This cancels the second order
pole from the Vandermonde determinant in the denominator so that (4.37) remains finite.
But by the permutation symmetry of∣∣H∨d×d
∣∣ and |σ1 · · ·σd|, (4.37) cannot have any poles
in any of the worldsheet coordinates. Again, a function of degree zero that has is glob-
ally holomorphic on a compact Riemann surface must be constant, so 〈Υ(σ1) · · ·Υ(σd)〉 is
completely independent of the insertion points, as expected for picture changing operators.
Above we obtained a representation of det′(H∨) in which we computed a minor involv-
ing the same rows and columns. Once again, we can obtain more general representations,
in which we compute arbitrary minors of H∨, by inserting picture changing operators for
the two flavours (µ1, µ2) of antighost at independent locations. That is, we replace
Υ(σ)→ Υ(σ, σ′) ≡ δ(µ1)〈ρ1, Z〉(σ) × δ(µ2)〈ρ2, Z〉(σ′) (4.40)
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JHEP04(2020)047
to compute a minor of H∨ from rows and columns corresponding to the independently
chosen insertion points of δ(µ1) and δ(µ2). As before, since det′(H∨) is competely permu-
tation symmetric in — indeed, completely independent of — all insertion points, there is
no real difference between any of these cases, although a judicious choice may help simplify
some calculations.
The conjugate Hodges matrix appeared to be the most complicated ingredient in the
gravitational scattering matrix as presented in [32]. Quite remarkably, it has turned out
to be one of the simplest.
4.4 The tree-level S-matrix
Combining the correlation functions (4.25) & (4.37) with the remaining integral over the
zero modes of the Y Z system — i.e., the space of holomorphic maps Z : Σ → PT — and
dividing by vol(GL(2;C)) to account for the zero modes of the ghosts associated to the
worldsheet gauge theory, we have found that⟨d+2∏i=1
∫Σδ2(γ)hi(Z)
n∏j=d+3
∫X
d1|2zHj(Z)d∏k=1
Υk
⟩
=
∫d4(d+1)|8(d+1)Z
vol(GL(2;C))det′(H) det′(H∨)
n∏i=1
∫Σhi(Z(σi))(σidσi) . (4.41)
Recalling the genus zero relation d = k+1 between the degree of the map and the NkMHV
level, this correlation function is exactly Mn,k as defined in (1.5). Summing over all d ≥ 0
and allowing all n ≥ 3 yields the complete tree-level S-matrix of N = 8 supergravity. The
ability to reproduce this formula for the complete classical S-matrix is a highly non-trivial
test of our claim that the worldsheet model proposed in section 3 does indeed describe
N = 8 supergravity.
5 Discussion
We have shown that the worldsheet theory defined by the action (3.32) and BRST opera-
tor (3.24) provides a twistor string description of N = 8 supergravity. The model depends
on a choice of infinity twistor, and different choices lead to N = 8 supergravity on flat or
curved space-times, with the R-symmetry gauged or ungauged. We showed that in the flat
space limit, g = 0 worldsheet correlation functions in this theory generate the complete
classical S-matrix of N = 8 supergravity, in the form discovered in [32] and proved to be
correct in [33]. By interpreting N = 8 supergravity as a twistor string, the present work
supplies the theoretical framework to explain why this form for gravitational scattering
amplitudes exists.
The ideas presented here suggest many avenues for further exploration. Let us conclude
by discussing some of these.
5.1 The SL(2;C) system
The most immediately important issue is to properly understand the role of the worldsheet
gauge theory. In the present paper, our primary concern was to reproduce the tree- level
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JHEP04(2020)047
S-matrix (1.5). At g = 0, the rank 2 bundle C2 ⊗ L is uniquely determined by degree
of L. Consequently, the holomorphic GL(2;C) bundle has no moduli at g = 0 and the
antighost M has no zero modes. We were able to account for the zero modes of N rather
naively, taking the quotient of the zero mode path integral by the obvious GL(2; C) action,
or ‘dividing by vol(GL(2))’.
At higher genus, holomorphic bundles do have a non-trivial moduli space even for a
fixed curve Σ, and this moduli space has been extensively investigated from many points
of view in both the mathematics [95–99] and physics literature [100–102]. In particular,
when g ≥ 2 the moduli space N of stable holomorphic SL(2;C) bundles has dimension
3g − 3, while that of stable GL(2;C) bundles has dimension (3g − 3) + g with the extra
g corresponding to the Picard variety of L. A dense open set of N may be identified
with the Teichmuller space of Σ. We have repeatedly mentioned that our twistor string
does not involve worldsheet gravity and so its path integral does not automatically include
an integral over the moduli space of Riemann surfaces. Nonetheless, it does know about
Teichmuller space via the moduli space of the rank 2 gauge bundle associated to D. In
fact, there is even a natural isomorphism between H1(Σ, TΣ) and H1(N, TN) (see e.g. [95,
98, 102]), so that deformations of the complex structure of Σ and of the SL(2) bundle are
to some extent interchangeable. The mechanism by which this is realized in the current
context, and the implications for the twistor string, cry out for a better understanding.
A closely related issue is the apparent absence of vertex operators inserted at a fixed
point p ∈ X, rather than on a fixed section Σ → X. A proper understanding of the
SL(2;C) system should include an operator which creates a puncture on Σ to which
our vertex operator is attached. Including such operators should amount to allowing
(parabolically stable [105]) holomorphic SL(2;C) bundles that have simple poles at pi ∈ Σ,
such that the monodromy of the associated flat connection is in a fixed conjugacy class
G ⊂ SL(2;C). With n such punctures, the moduli space of such meromorphic bundles has
dimension 3g− 3 + n.
As an obvious application, vertex operators associated to punctures on X are likely to
be important if one wishes to have a worldsheet description of factorization [55, 78]. The
formula (4) for the gravitational scattering amplitudes was shown to obey the expected
factorization properties in [33]. However, the derivation given there was rather involved,
because by necessity it dealt with the path integral after integrating out everything but
the Z zero-modes. By working directly with the vertex operators, one should be able to
provide a simpler proof (following the general pattern in string theory), as the terms that
may become singular in the factorization limit are isolated more cleanly.
5.2 Higher genus
The discussion of section 5.1 has an immediate corollary that perhaps bears some relation
to the debate about whether N = 8 supergravity could be perturbatively finite [106–112].
Usually, string theory is UV finite because the worldsheet theory is modular invariant. We
do not integrate over Teichmuller space, but rather over its quotient by the mapping class
group. This renders harmless any potential divergence as Im(τ)→ 0, and this potentially
dangerous region becomes (real) codimension 2 rather than codimension 1. In the theory
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JHEP04(2020)047
studied here though, worldsheet gravity is replaced by a worldsheet gauge theory. As
mentioned above, we expect the path integral to involve an integral over the moduli space
of stable SL(2;C) bundles, not the moduli space of curves. Although many aspects of the
gauge theory (such as the symplectic form) are invariant under the mapping class group, it
is not clear that we should really expect to take the quotient by modular transformations.
If not, then the potentially dangerous region Im(τ) → 0 is still present. Of course, it is
perfectly possible that the integrand still happens to have no singularity here — and indeed
we should expect this at low genus — but this requires calculation.28 We are unable to
offer the usual string theoretic guarantee that there is simply no place for UV divergences
to arise.
Whatever the fate of N = 8 supergravity at higher loops, the current consensus is
that we do not expect any UV divergences when g < 7 [106–112]. What prospect does
the twistor string have for computing these ‘intermediate’ loop amplitudes? Hopefully, the
above discussion has made it clear that we cannot give a proper answer to this question
without first understanding the role of the worldsheet gauge theory. Nevertheless it is
clear that many properties of these amplitudes are correctly reflected by the worldsheet
theory. In particular, the zero modes of the βγ- and µν-systems will yield higher loop
Hodges matrices that have the correct dependence on the infinity twistors [ , ] and 〈 , 〉required by factorization (at least in the generic case with d > 2 − 2g; in general we
would need to work with n ‘fixed’ vertex operators and an appropriate number of picture
changing operators for the βγ-system). The factors of (ij) appearing in these matrices
at g = 0 naturally generalize to the appropriate Szego kernels at higher genus, while the
Vandermonde determinants coming from the correlation function of insertions fixing the
zero-modes will involve a basis of holomorphic sections of L over a genus g curve. All these
ingredients can be written in terms of (higher-order) theta functions. See [113] for a related
discussion in the context of the original twistor string models.
Even if successful, it is doubtful that the twistor string would reproduce even one-loop
amplitudes in a form that permits direct comparison with known results in the litera-
ture [28] (though some of the expressions found in [114–116] may be closer). A direct
assault on the resulting integrals is unlikely to be successful; the integrals over the mod-
uli space of higher degree twistor curves is challenging even at g = 0 [117]. The most
promising approach is probably to check that the resulting expressions have all the correct
factorization properties.
5.3 Boundary correlation functions in AdS4
In this paper, we concentrated on taking the flat space limit so as to extract gravitational
scattering amplitudes and make contact with the known literature. However, the theory is
equally capable of describing supergravity or gauged supergravity on AdS backgrounds —
we simply keep the infinity twistor or infinity supertwistor non-degenerate.
28Another intriguing but very speculative idea would be that the theory allows us to take the quotient
by a g-dependent congruence subgroup of the mapping class group that becomes trivial when g is greater
than some minimum value g0, signalling the onset of UV divergences.
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JHEP04(2020)047
On anti de Sitter space, the natural observables are not scattering amplitudes but
rather boundary correlation functions. These are obtained by choosing the external wave-
functions to represent bulk-to-boundary propagators, that is, solutions of the free equations
of motion on AdS, with a singularity on the conformal boundary. Such wavefunctions have
a very simple description in twistor space, known in the twistor literature as ‘elementary
states’ (see e.g. [118]). For example, consider the elementary state
φ(Z) =1
A·Z B·Z∈ H1(PT,O(−2)) (5.1)
representing a scalar field in twistor space. If the line AB is chosen to obey 〈A,B〉 = 0,
then it lies at infinity. In particular, if I is the non-degenerate infinity twistor associated
to AdS4, then this twistor line represents a point y on the thee dimensional conformal
boundary. Using the standard incidence relation µα = xααλα, the Penrose transform
of (5.1) appropriate to AdS4 is
K(x, y) =
∮〈Z, dZ〉
A·Z B·Z=
∮(1 + Λx2)〈λdλ〉
(Aαxαα + Aα)λα (Bβxββ + Bβ)λβ
∝ (1 + Λx2)
(x− y)2
(5.2)
where we used the non-degenerate infinity twistor in the measure 〈Z, dZ〉. This is the bulk
to boundary propagator for a scalar field, written in the coordinates where
ds2 =dxµdxµ
(1 + Λx2)2(5.3)
is the AdS4 metric and where (x− y)2 is computed using the flat metric.
Using states such as (5.1), it should be possible to use the formalism of this paper to
compute arbitrary n-point boundary correlators, again in the form of an integral over the
moduli space of degree d curves in CP3, at least at g = 0. One obvious feature is that the
(n−d−2)× (n−d−2) worldsheet Hodges’ matrix and the d×d conjugate Hodges’ matrix
will combine into a single (n− 2)× (n− 2) worldsheet matrix, with the off-block-diagonal
terms being proportional to the cosmological constant. These terms arise because with
a non-degenerate infinity twistor, both the ρρ- and Y Z-systems have cross-contractions
between the Oh vertex operators representing the external states and the picture changing
operators Υ. Indeed, one could anticipate this happening. The generalization of Hodges’
MHV amplitude to the worldsheet Hodges’ matrices was deduced [32] starting from the
observation that factorization requires the n-particle Nd−1MHV flat space tree amplitude
to contain n− d− 2 powers of [ , ] and d powers of 〈 , 〉 when written in twistor space. But
with a non-degenerate infinity twistor these two objects are really equivalent.
In the twistor string, as in usual string theory, factorization of scattering amplitudes is
closely related to collision of vertex operators on the worldsheet [33, 36, 39]. Factorization
of boundary correlators in AdS has been investigated recently in [119–122], where it is
shown that (tree-level) Witten diagrams in AdS obey a natural generalization of BCFW
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JHEP04(2020)047
recursion. An important observation related to this is that the structure
M(Z1, . . . , Zn) =∑∫
D3|8Z ∧ dt
t∧ML(Z1 + tZn, . . . , Z) ∧MR(Z, . . . , Zn) (5.4)
of the BCFW recursion relation in twistor space [37, 38] is completely conformally invariant
when expressed in terms of external ‘twistor eigenstates’
hi(Z) = δ3|8
(Z,Zi) ≡∫
ds
s3∧ δ4|8
(Zi + sZ) . (5.5)
In particular, the infinity twistor arises only via the three-point functions that seed the
recursion relation. We can obtain AdS boundary correlators by integrating (5.4) against
appropriate boundary elementary states hi(Zi). Thus, in twistor space, the BCFW recur-
sion for Witten diagrams in AdS is exactly the same as BCFW recursion for flat space
amplitudes. Only the translation back to momentum space (associated to the boundary
∂AdS) and the three-point functions are different. It would be fascinating to relate these
observations to the structures of Witten diagrams found in [119–122]. Of course we are
limited to the case that the bulk AdS space is four-dimensional.
Finally, via analytic continuation to dS, boundary correlators of gravitational modes on
AdS may even have cosmological applications [123]. The ideas presented here may provide
a way to extend the calculations of [123] to higher-point functions. The (n > 3)-point
gravitational wave power spectrum is admittedly a rather esoteric cosmological observable!
5.4 Other issues
We briefly mention various other issues.
Firstly, the theory we have presented is purely chiral really provides a top holomorphic
form on the moduli space. It is this ‘scattering form’ that was found in [32]. To recover
the actual scattering amplitudes we must still pick a 4d-dimensional29 real integration
cycle on which to integrate this form. When d = 1 and g = 0, the moduli space is
simply complexified space-time and the appropriate integration cycle is just a copy of
real Minkowski space. For higher degrees the appropriate contour is less easy to define.
One possibility, suggested in [12] and hardwired into Berkovits’ model [56], is to pick real
structures30 τ1 on Σ and τ2 on CP3 and ask that the map is equivariant in the sense that
Z τ1 = τ2 Z. In ultrahyperbolic space-time signature, these real structures fix an S1
equator on Σ at g = 0 and an RP3 real slice of twistor space. However, some care is needed
in the interpretation of wavefunctions on real twistor space (see e.g. [37] for a discussion).
Other integration cycles of interest include those that compute factorization channels of
amplitudes, ultimately yielding ‘leading singularities’. It would be good to know whether
the twistor string naturally picks a preferred integration cycle for us, or whether this is
additional data that must be specified.
29This is in the case that the wavefunctions are represented in terms of Dolbeault cohomology classes
H(0,1)(PT,O(2)). A description in terms of sheaf cohomology would require us to pick a (4d + n)-
dimensional cycle.30Recall that a real structure is an antiholormorphic involution squaring to the identity.
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JHEP04(2020)047
In this paper, although we identified the relevant transformations of X that were being
gauged, we did not attempt to write down a classical action theory that realized this gauge
symmetry off-shell. Instead, we moved right away to a gauge fixed model together with
its ghosts and BRST symmetry. It would be interesting to construct the unfixed model,
particularly is this would likely shed further light on the role of the GL(2)-system. Such a
model would appear to involve 2 charged gravitinos and 2 charged gauginos in addition to
the GL(2) gauge fields.
Next, the vertex operators Oh and Oh that we obtained are natural analogues of
Neveu-Schwarz sector vertex operators in the superstring. It is important to know what,
if anything, the Ramond sector could be in the present context. Unlike conformal gravity
modes in the original twistor string, we would not expect Ramond sector operators to be
generated at g = 0 if they are not present (pairwise) in the external states. If they exist,
their role at g ≥ 1 is clearly important to understand.
We saw in section 4.2.1 that, when restricted to constant maps, the string field theory
of our model is the twistor action for self-dual N = 8 supergravity found by [85]. The string
field theory of the full model should thus include a further term representing worldsheet
instanton contributions. Presumably, only the degree 1 instantons need be included, as
is the case in usual string theory [124] and as in the analogous twistor action for N = 4
super Yang-Mills [3]. In our context, these would represent off-shell gravitational MHV
vertices. This strongly suggests that despite the difficulties [22] with Risager recursion for
gravity [20, 21] an MHV formalism for gravity exists. It would clearly be of great interest
to find a twistor action for non self-dual gravity. The deformed worldsheet action (3.51)
perhaps provides a good starting-point. See [24] for an earlier attempt to construct a
twistor action for gravity. An important step in the right direction has recently been
taken in [40].
Last but not least, it would be very interesting to revisit the potential existence of a
twistor string for pure N = 4 super Yang-Mills in the light of this paper. One approach
might be to try to understand the meaning of the duality between colour and kinemat-
ics [125] in a twistor framework. This duality has certainly lead to great progress in the
computation of multi-loop gravitational amplitudes in momentum space, typically with
n = 4. The similarity between the twistor action (4.28) for self-dual gravity and (4.29) for
self-dual Yang-Mills is surely no coincidence. Yang-Mills amplitudes are completely per-
mutation symmetric in the external states provided we include their colour factor. Perhaps
they also admit a Hodges matrix form.
Acknowledgments
I am very grateful to N. Arkani-Hamed, P. Goddard, J. Maldacena, L. Mason, H. Verlinde
and E. Witten for helpful discussions. I am supported by an IBM Einstein Fellowship of
the Institute for Advanced Study.
A Some properties of algebraic βγ-systems
In this appendix we will compute some correlation functions of operators in βγ systems
that are ingredients in computing the twistor string worldsheet correlator (4.1). Nothing in
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JHEP04(2020)047
this appendix is new — all (and much more besides) may be found in [126] and in section
10 of [55], which we follow closely.
In constructing the twistor string theory, we imposed no reality conditions on the
worldsheet fields (see the discussion in section 5.4). Thus the path integral over these fields
should be understood as a formal algebraic operation. This is exactly the usual case for
Berezin integration of fermionic variables, and so the discussion of [55] proceeds by relating
integrals over bosonic fields to integrals over fermionic fields that are easier to understand.
Thus we consider the path integral over anticommuting fields that we call b and c with
action
Sbc =1
2π
∫Σb ∂c . (A.1)
The result of this path integral depends on the quantum numbers of these fields. Without
loss of generality, we can assume that
c ∈ Ω0(Σ, L) and b ∈ Ω0(Σ,KΣ ⊗ L−1) (A.2)
for some line bundle L, and we assume the ∂-operator in the action in (A.1) acts appropri-
ately on sections of L. Then zero modes of c are globally holomorphic sections of L while
zero modes of b are globally holomorphic sections of KΣ ⊗ L−1. By Serre duality, this is
H1(Σ, L). In the case that L is a spin bundle, so that L2 = KΣ, (generically) neither field
has zero modes and the bc path integral is∫D(b, c) exp
(− 1
2π
∫Σb ∂c
)= det(∂
K1/2Σ
) , (A.3)
or in other words the determinant of the Dirac operator on Σ. As explained in [127, 128]
this may be written in terms of the Riemann theta functions associated to Σ and the choice
of spin structure. When g = 0, we may take it to be a constant.
For any other choice of L, at least one of b or c will have zero modes. By the usual
rule∫
dθ · 1 = 0 of Berezin integration, the path integral (A.3) vanishes. To obtain a
non-vanishing result, we must insert exactly enough fields to absorb the zero modes. For
simplicity, let us suppose that c has some number m of zero modes, so that we may expand
it as
c(z) =
m∑i=1
ciYi(z) + non zero-modes (A.4)
where ci are anticommuting constants and the Yi form a basis of H0(Σ, L) (here written in
terms of a local coordinate z ∈ U ⊂ Σ). With m insertions of c, the path integral becomes∫D(b, c) c(z1) · · · c(zm) exp
(− 1
2π
∫Σb ∂c
)= det′(∂L)×
∫ m∏i=1
dci c0(z1) · · · c0(zm) (A.5)
where c0(z) =∑
i ciYi(z), and where the determinant is provided by the path integral
over the non-zero modes of the bc-system. Since the c’s anticommute, (A.5) must be
antisymmetric under the exchange of any pair of insertion points zi and zj . It must also
be holomorphic in all of these insertion points. Thus we find∫D(b, c) c(z1) · · · c(zm) exp
(− 1
2π
∫Σb ∂c
)= det′(∂L)× det(Y) , (A.6)
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JHEP04(2020)047
where Y is the m ×m matrix with entries Yij = Yi(zj). Equation (A.6) is the standard
result for fermions.
In Berezin integration, if c(z) is fermionic then δ(c(z)) = c(z). In addition, because
eτ = 1 + τ if τ2 = 0, we can represent δ(c(z)) in integral form as
δ(c(z)) =
∫dτ exp (τc(z)) (A.7)
where τ is an auxiliary anticommuting variable. This is clearly analogous to the usual
integral representation of the Dirac δ-function. Following [55] we thus introduce m =
h0(Σ, L) such constant anticommuting variables (τ1, . . . , τm) and let b indicate the collection
of fields (b(z); τ1, . . . , τm). We also introduce the extended action
Sbc
=1
2π
∫Σb ∂c −
m∑i=1
τic(zi) (A.8)
and the extended path integral measure
D(b, c) = D(b, c)n.z.m dc1 · · · dcm dτ1 · · · dτm , (A.9)
where D(b, c)n.z.m. is the measure on the infinite dimensional space of non zero-modes.
Combining (A.7)–(A.9) we see that the path integral (A.6) may be rewritten as
det′(∂L)× det(Y) =
∫D(b, c) e−Sbc
m∏i=1
δ(c(zi)) =
∫D(b, c) e−S
bc (A.10)
in terms of the extended set of fields and action. The virtue of thinking about (A.6) in
this way is that we have changed a path integral with insertions into a simple path integral
over a Gaussian action.
It is now straightforward to understand the bosonic case that is actually needed in
section 4. Suppose β and γ are fields on Σ with exactly the same quantum numbers as
b and c, except that they are commuting fields. In the case that L2 = KΣ, in contrast
to (A.3) we have ∫D(β, γ) exp
(− 1
2π
∫Σβ ∂γ
)=
1
det(∂L)(A.11)
giving the inverse of the determinant, as is familiar from Gaussian integration.31 When
L is a more general line bundle such that γ has zero modes, the path integral diverges
(or, without a reality condition, is ill-defined) because of the integration over these zero
modes. They can be fixed by inserting δ-function operators, and again we represent these
in integral form as
δ(γ(z)) =
∫dt exp (tγ(z)) . (A.12)
31With no reality condition on the βγ-system, this is really a definition of what we mean by the Gaussian
path integral. See sections 3 & 10 of [55].
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JHEP04(2020)047
Constructing an extended action and path integral measure as before, but now with com-
muting variables, we again convert multiple insertions of such δ-function operators into a
Gaussian integral. We thus find∫D(β, γ) exp
(− 1
2π
∫Σβ ∂γ
) m∏i=1
δ(γ(zi)) =1
det′(∂L) det(Y)(A.13)
where Y is the same matrix of zero modes as before. Notice that with our formal algebraic
treatment, there is no modulus sign on the determinants on the right hand side. Notice
also that if γ(z) represents a section of L, then δ(γ(z)) should transform as a section of
L−1. Both sides of (A.13) transform as sections of ⊗iL−1|zi .In the main text, we will be interested in the cases L = K
−1/2Σ ⊗L and L = K
1/2Σ ⊗L,
where L is a line bundle of degree d. In particular, when g = 0, L is uniquely determined
to be OCP1(d). The appropriate zero modes are then
Yi(σ) =σα1 · · ·σαd+1
(σdσ)1/2for K
−1/2Σ ⊗ L
Yi(σ) = σα1 · · ·σαd−1 (σdσ)1/2 for K+1/2Σ ⊗ L
(A.14)
where i runs over all possible choices of the indices α1, . . . , αd+1 or α1, . . . , αd−1, respec-
tively. Inserting these zero modes into Y in (A.13) gives equation (4.13) for the zero modes
of each flavour of the worldsheet fields γaa, and (4.31) for the zero modes of each copy of
the worldsheet field µa. Recall that det′(∂L) is a constant at g = 0.
Far more information about correlation functions in algebraic βγ systems can be
found in [55].
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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